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Chapter 14: p-Permutation Modules
Chapter 14
p-Permutation Modules In this chapter, we provide a thourough investigation of p-permutation modules. The coefficient ring R is chosen to be either a field F of prime characteristic p or a complete discrete valuation ring such that c h a r ( R / J ( R ) )= p. Although a given RG-module V need not be a permutation module, it is quite possible that the restriction Vp of V to RP, where P is a Sylow psubgroup of G, is a permutation module. Such an RG-module V is called a ppermutation module. It turns out that the p-permutation modules are precisely direct summands of permutation modules. The chapter is divided into six sections. In Sec.1, we introduce relevant definitions and provide various characterizations of p-permutation modules. In particular, we show that an indecomposable RG-module is a ppermutation module if and only if it has a trivial source. We then demonstrate that any ppermutation FG-module lifts uniquely to a p-permutation RG-module, and show that the corresponding residue class map is surjective. In Sec.2, we provide further characterizations of p-permutation modules via the notions of monomial and virtually monomial modules. The corresponding results are extracted from a work of Dress (1975). The section also contains some additional results of independent interest. In Sec.3, we introduce an important tool for the study of p-permutation modules, namely the Brauer morphism. Most of the results presented can be found in Brouk (1985) and Brouk and Puig (1980). The corresponding theory is then applied in Sec.4 for the study of Scott modules. In Sec.5, we provide a detailed investigation of vertices of p-permutation modules. Many properties of this nature were investigated in a work of Klyachko (1979) and Brouk (1985). One of these properties asserts that, 771
772
p-Permutation Modules
for any psubgroup P of G , the number of nonisomorphic indecomposable p-permutation RG-modules with vertex P is equal to the number of Fconjugacy classes of pregular elements of N G ( P ) / P . In the final section, Sec.6, we prove an interesting result due to Puig which asserts that the NG(P)-algebras ( E n d F ( V ) ) ( P )and E n d F ( V ( P ) )are identifiable. Here V is a p-permutation FG-module and P is a p-subgroup of G.
1
Definitions and basic properties
In this section, G denotes a finite group and R is either a field of characteristic p > 0 or a complete discrete valuation ring such that c h a r ( R / J ( R ) )= p > 0. In what follows, we put F = R/J(R). If H is a subgroup of G, then all RHmodules are assumed to be finitely generated. We say that an RG-module V is a p-permutation m o d u l e if the restriction V p of V to a Sylow psubgroup P of G is a permutation RP-module. Thus a nonzero RG-module V is a p-permutation module if and only if V is R-free of finite rank with an R-basis on which a Sylow p-subgroup of G acts as a permutation group. Our point of departure is the following basic result which can be found in Dress (1975).
T h e o r e m 1.1. Let V = V1 @ ... @ V, be a direct decomposition of a nonzero RG-module V into indecomposable submodules. Then the followang conditions are equivalent : (i) V is a p-permutation module. (ii) For any p-subgroup Q of G , VQ is a permutation RQ-module. (iii) V is a direct summand of a permutation module. (iv) There exist subgroups H I , . . . ,H , of G such that, for a n y i E { 1, . . . ,n } , V, is isomorphic to a direct summand of ( 1 ~ ~ ) ~ . (v) Each V , is a p-permutation module. (vi) Each V , has trivial source. Proof. (i) 3 (ii) : Let P be a Sylow p-subgroup of G such that Vp is a permutation RP-module. Then Q C gPg-' for some g E G. If vl,. . . ,Vk is a permutation basis for V p , then g q , . . , ,gvk is a permutation basis for Vgpg-l and hence for VQ,as required. (ii) + (iii) : Let P denote a Sylow p-subgroup of G. By assumption, V p
1 Definitions and basic properties
773
is a permutation RP-module. Hence (Vp)' is a permutation RG-module. But V is P-projective, so V is isomorphic t o a direct summand of ( V p ) G , proving (iii). (iii) + (iv) : By hypothesis, there is a permutation RG-module M such t h a t each V, is a direct summand of M . Because V, is indecomposable, it follows from the unique decomposition property that V, is isomorphic to a direct summand of ( 1 ~ ~for ) ' some subgroup Hi of G, 1 5 i 5 n , proving property (iv). (iv) (v) : Because ( 1 ~ ~ is ) ' a permutation module, it is also a ppermutation module. Hence we need only show that a nonzero direct summand N o€a ppermutation module M is a ppermutation module. If P is a Sylow psubgroup of G such that Mp is a permutation RPmodule, then N p is a direct summand of M p . But Mp is a direct sum of modules of the form ( l ~ ) ' ,where Q is a p-subgroup of G. By Lemma 11.1.4, ( 1 ~ ) 'is indecomposable. Hence, by the unique decomposition property, N p is a direct sum of modules of the form ( 1 ~ )Thus ~ . N is a p-permutation module. (v) + (vi) : Put M = V, and let P be a Sylow p-subgroup of G such that Mp is a permutation RP-module. If Q P is a vertex of M , then M is Q-projective and so M is isomorphic t o a direct summand of (MQ)'. By hypothesis, MQ is a direct sum of modules of the form (ls)Q,where 5' is a subgroup of Q. Hence M is a direct summand of (1s)Q for some subgroup S of Q. Because M is S-projective and S Q , S is a vertex of M and hence M has trivial source. (vi) + (i) : It suffices t o verify that each V, is a ppermutation module. By the implication (i) (iii), V, is a direct summand of a permutation (hence p-permutation) module. Hence, by the proof of the implication (iv) + (v), V, is a p-permutation module. H
*
*
As an easy consequence, we now derive
Corollary 1.2. Let H be a subgroup o f G and let V and W be RGmodules. (i) If V = VI @ . . @ V, for some nonzero RG-modules V,, then V is a p-permutation module if and only if each I$ is a p-permutation module. (ii) If V and W are p-permutation modules, then so is V @R W . (iii) If V is a p-permutation module, then VH is a p-permutation module. (iv) If U is a p-permutation module RH-module, fhen UG is a p-permutation
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RG-module. (v) If V is a p-permutation module, then so is V'.
Proof. (i) Direct consequence of Theorem 1.1. (ii) If {v;} and {wj} are R-bases of V and W , respectively, which are permuted by a Sylow psubgroup of G, then the same is true for the R-basis {v; 8 wj} of V 8~W . (iii) If Q is a Sylow psubgroup of H , then VQ= (VH)Q is a permutation RQ-module, by virtue of Theorem l.l(ii). (iv) This is obvious. (v) Let P be a Sylow psubgroup of G. Then Vp is a permutation RPmodule. Since (Vp)* Z ( V * ) p , it follows from Lemma 11.1.6 that (V*)pis a permutation RP-module. Thus V* is a ppermutation module. W Let U , V be two RG-lattices. Then, for any f E HomRG(U,V),
Hence f induces an FG-homomorphism
given by
-
Here 21 is the image of u in 0 = U/J(R)U and f ( u ) is the image of f ( u ) in V = V / J ( R ) V .We remind the reader that the additive homomorphism
is called the residue class map . Owing to Theorem 2.7.11, the residue class map induces an injective F-linear map
v)
, lift The image of this map consists precisely of all X E H o ~ F G ( ~ which to H o ~ R G ( UV, ) . We now apply the above information to the special case where U and V are permutation RG-modules.
1 Definitions and basic properties
775
Lemma 1.3. Let U and V be permutation RG-modules.
Then the
residue class map induces an F-isomorphism
If U = V , then the above is an isomorphism of F-algebras. Proof. Since a permutation module is a direct sum of transitive permutation modules, we may assume that both U and V are transitive permutation modules. Then U 2 ( l n ~ ) 'and V 2 (1~s)'for some subgroups H and S of G. Hence 2 ( ~ F H and ) ~ =" ( 1 ~ s ) It ~ .follows from Lemma 11.1.8 that both sides in (1) have the same F-dimension (which is equal t o the number of (S,H)-double cosets in G). This clearly implies the first
v
assertion. The second assertion being obvious, the result follows.
We are now ready t o record the following important property of ppermutation modules.
Theorem 1.4. The following properties hold : (i) Any p-permutation FG-module lifts uniquely (up to isomorphism) to a p-permutation RG-module. (ii) For any p-permutation RG-modules U and V , the residue class map induces an F-isomorphism
If U = V , then the above is an isomorphism of F-algebras.
Proof. (i) Let M be a p-permutation FG-module. Then, by Theorem l.l(iii), M is a direct summand of a permutation FG-module, say N . Let E E E ~ ~ FN G ) be ( the corresponding idempotent, and choose any permutation RG-module U with U = N . Owing t o Lemma 1.3 and Theorem 1.6.20 in Vol.1, E can be lifted t o an idempotent, say e , of EndRG(U). Setting V = e ( U ) it follows that V is a direct summand of U (hence V is a ppermutation module) such that
v 2 E(N) = M Thus M lifts t o a p-permutation RG-module. Suppose that Ul and V, are p-permutation RG-modules with UI 2 V I .
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We may assume that U1 and V1 are direct summands of the same permutation RG-module U . Choose idempotents e , f of EndRG(U) with e ( U ) = U1 and f ( U ) = V1. Since 01 2 we have e( U ) 2 $( U ) . Hence, by Proposition 1.9.4(iii) in Vol.1, EndFG( U ) Z EndFG( U)f
q,
Therefore, by Lemma l.S(with V = U ) and Theorem 1.6.20 in Vol.1,
EndRG( u ) e !X EndRG( u )f Again, by Proposition 1.9.4(iii) in Vol.1, U1 2 Vl and (i) is established. (ii) Since a ppermutation module is a direct summand of a permutation module, the required assertion is a consequence of Lemma 1.3. As an easy application of the above, we next record the following result.
Theorem 1.5. Let U , V be p-permutation RG-modules. Then (i) U E V if and only i f 2 (ii) U is indecomposable i f and only if U is indecomposable. (iii) If U is indecomposable with vertex Q , then 0 is indecomposable with vertex Q .
u v.
Proof. (i) This is a direct consequence of Theorem 1.4(i). (ii) It is clear that if U is indecomposable, then so is U . Conversely, assume that U is indecomposable. If U = X @ Y for some nonzero FGmodules X and Y , then X 2 and Y = U 2 for some p-permutation RG-modules U1 and U2 (see Theorem 1.4(i)). Then U E UI @ U2 and, by (i) U 2 U1 @ U2. This is a contradiction, since U1 # 0 and U2 # 0. Thus U is indecomposable and the required assertion follows. (iii) Let P be a vertex of U . We must show that P and Q are G-conjugate. Since U is a component of ( U Q ) ~it,follows that U is a component of ( 0 ~ ) ~ . Hence P is G-conjugate t o a subgroup of Q. On the other hand, there exists f E E n d F p ( u ) such that 10 = T r s ( f ) . By Theorem 1.4(ii), we may write f = 1for some X E E n d R p ( U ) . Thus
ul
lu which forces
E T$(X)
(modJ(R)EndRG(U))
2 Dress’s theorems
777
This shows that U is P-projective and so Q is G-conjugate t o a subgroup of P . Thus P and Q are G-conjugate, as required. Let a(RG) be the Green ring (see Sec.1 of Chapter 10). Denote by a,(RG) the Z-linear span of all [ V ] where , V is a p-permutation RG-module. Then, by Corollary 1.2(ii), a,(RG) is a subring of a(RG).
Corollary 1.6.
The map
is a ring isomorphism. Proof. It is clear that a,(RG) is a free Z-module freely generated by all [ V ] ,where V runs through the nonisomorphic indecomposable p-permutation RG-modules. A similar statement holds for the subring a,(FG) of a(FG). Hence the desired conclusion is a consequence of Theorem 1.5.
2
Dress’s theorems
Throughout this section, we fix a field F of charactristic p > 0 and a finite group G. If H is a subgroup of G, then any FH-module is assumed to be finitely generated. Our aim is t o provide some further characerizations of p-permutation modules which are contained in a work of Dress (1975). This will be achieved with the aid of the notion of virtually monomial modules defined below. Let V be an FG-module. Then V is said to be monomial if there exists a basis q , . . . ,TJof , V such that G permutes the subspaces F q , . . . ,Fv, of V . Expressed otherwise, V is monomial if and only if V is a direct sum of modules of the form W G ,where W is one-dimensional FH-module for some subgroup H of G. Following Dress (1975), we say that V is virtually monomial if there exist monomial FG-modules V1 and V2 such that
v 69 v,?z v, It is plain that the direct sums of monomial (virtually monomial) modules are again monomial (virtually monomial) modules. Our main result of this section will be proved with the aid of the following theorem which is of independent interest.
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Theorem 2.1. (Dress (1975)). Let F be an algebraically closed field of characteristic p > 0 and let V be an indecomposable FG-module with vertex Q and a trivial source. Then there exist subgroups H I , . . . ,H, of G and one-dimensional FH;-module Q, 1 5 i 5 n, such that : @ v,G (for some IC < n). (i) v @ V? @ .. @ v,G 2 @ (ii) Each H i , 1 5 i 5 n, has a normal Sylow p-subgroup P; such that H;/P; is elementary (i.e. a direct product of a cyclic group and a q-group for some prime q ) and such that
vgl
g;P;gzT1C Q for some g; E G
(1 5 i
5 n)
Proof. We apply induction on IQI. First suppose that Q = 1 and let {E;liE I } be the set of elementary subgroups of G. By Corollary 3.5.13, for each i E I , there exist FE;-modules M ; and N ; such that 1~ @ (@iEzM;G) and
Nf
(1)
have the same composition factors (counting multiplicities). Taking into account that for any FG-module W , the FG-module V @ 3W~ is projective, we may tensor both modules in (1) with V t o obtain the following isomorphism of FG-modules :
v @ (@iEI(VEi @ F MilG) 2 @i€Z(VE,@ F NilG Thus, because V E @F ~ M ; and VE,@F N; are projective FE;-modules, we may harmlessly assume that G is an elementary group. If p { ]GI, then there is nothing to prove. We may therefore assume that pl IGl and write G = P x H , where P is a pgroup and H is an elementary p’-group. Let Vl,.. . ,& be all nonisomorphic simple FH-modules. Then it is clear that..,:fI .,KG are all nonisomorphic projective indecomposable FG-modules. Since V is projective, we have V 2 yGfor some i E (1,. .. ,t} and we are again reduced to the case where p { [GI. This establishes the case where Q = 1. Now suppose that IQI > 1 and let H = NG(Q). By the Green correspondence, there exists an indecomposable FH-module W with the same vertex Q and trivial source such that
W G E Uo @ UI @ . . . @ Urn
(U; is indecomposable, 0 5 i 5 m )
with Uo 2 V and IQI < IQil, Q; is a vertex of U; for i E (1,. . . ,m}. Bearing in mind that W is a direct summand of a permutation FH-module, W Gis a
2 Dress’s theorems
779
direct summand of a permutation FG-module. Thus U1,.. . , U,,, are direct summands of permutation FG-modules. Hence, by Theorem 1.1 and the induction hypothesis, the theorem holds for each FG-module U;, 1 5 i 5 m. Consequently, we need only prove the result for W , i.e. we may harmlessly assume that Q d G. Because Q Q G, we see that Q acts trivially on ( 1 ~and ) ~ hence on its direct summand V . Therefore V is inflated from an F(G/Q)-module V , where V is also projective. Consequently, we may apply the case Q = 1 t o infer that V satisfies the conclusion of the theorem (with G replaced by G/Q). This means that there exist subgroups
Hl/Q,...,Hn/Q of G/Q and one-dimensional F(Hi/Q)-modules q, 1 5 i 5 n, such that @ - .. CEI V: (L < n). (a) V @ V p -..@ V f 2 (b) Each H;/Q, 1 5 i 5 n , is elementary of order not divisible by p. Hence, if V , is the FG-module inflated from q,then by (a) and (b), H I , . . . , H , and V,, . . . ,V, satisfy the conclusion of the theorem. This concludes the proof of the theorem. W
Vzl
Specializing to projective FG-modules, we immediately deduce Corollary 2.2. (Dress (1975)). Let F be an algebraically closedfield of characteristic p > 0 and let V be a projective FG-module. Then there exist subgroups HI,. .. ,H , of G and one-dimensional FH;-module V,, 1 5 i 5 n , such that
v @ V? @ - .. CB v,G
2
vLl
..
@
v,G (L < n )
and such that each H i is an elementary p’-group. Proof. We may clearly assume that V is indecomposable, in which case Q = 1 is the vertex of V . Hence the desired conclusion follows from Theorem 2.1. W Another consequence of Theorem 2.1 is given by Corollary 2.3. (Dress (1975)). Let F be an algebraically closed field of characteristic p > 0 . Then there exist subgroups H I , . . . ,If, of G and one-dimensional FH;-module V;, 1 5 i 5 n, such that
p-Permutation Modules
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--
k+l @ @ V: (for some k < n). (i) 1~@ KG @ * @ VkG - VG (ii) Each H;, 1 5 i 5 n, has a normal Sylow p-subgroup P; such that H;/P; is elementary.
Proof. Let Q be a Sylow p-subgroup of G. Then, by Theorem 6.3.7, Q is a vertex of 1 ~It. is clear that 1~ is a source of 1 ~Hence . the desired assertion follows by virtue of Theorem 2.1. H
The result above allows us t o deduce the following additional property
Theorem 2.4. (Dress (1975)). Let F be a n algebraically closed field of characteristic p > 0. Then, for any FG-module V , there exist subgroups H I , . . . ,H , of G and FH;-modules W;, 1 5 i 5 n, such that : (i) V @ W y @ @ W f 2 WkG tl @ @ W: (for some k < n). (ii) Each H i , 1 5 i 5 n , has a normal Sylow p-subgroup with elementary
---
---
factor group.
Proof. Tensoring the isomorphism in Corollary 2.3(i) with V , we ob-
tain
v @ (@%,(v@F’ yG))
@y=k+l(V@ p KG)
Bearing in mind that, by Theorem 18.5.1 in Vol.1,
the result follows by setting W; = VH~@ F
K, 1 5 i 5 n. W
We have now come the the demonstration for which this section has been developed.
Theorem 2.5. (Dress (1975)). Let F be an arbitrary field of characteristic p > 0 and let V be a nonzero FG-module. Then the following conditions are equivalent : (i) V is a p-permutation module. (ii) V is a direct summand of a monomial FG-module. Moreover, under the assumption that F is algebraically closed, each of the above conditions is equivalent to : (iii) V is virtually monomial.
3 The Brauer morphism
78 1
Proof. Because each permutation FG-module is monomial, the implication (i) + (ii) is a consequence of Theorem 1.1. The implication (iii) + (ii) is a consequence of the definition of virtually monomial modules. Observe also that if F is algebraically closed, then (i) implies (iii) by virtue of Theorems 1.1 and 2.1. By the foregoing we need only show that (ii) implies (i) for an arbitrary F . To do this, we may, by Theorem 1.1, assume that V 2 W G where W is a one-dimensional FH-module for some subgroup H of G. But W is a p-permutation module because a Sylow p-subgroup of H acts trivially on W . Thus, by Corollary 1.2(iv), V is a p-permutation module. This concludes the proof of the theorem.
3
The Brauer morphism
An important tool for the investigation of p-permutation modules is the so called Brauer morphism. Our principal goal here is to record some basic properties of the Brauer morphism with an eye to future applications in the next section. Most of the results recorded are contained in Broud (1985) and Broub and Puig (1980). As usual, we begin by introducing the notation and assumptions. Throughout, p denotes a prime number and R a commutative local ring with
c h a r ( R / J ( R ) )= p We put F = R / J ( R ) and fix a subgroup K of a finite group G. Let V be an RG-module. Given a subgroup H of li, we put
v,l(= Trjy(lnv(VH)) and
V (K ) = l n v ( V ~ ) /
VE
+ J(R)lnv(V~)
where H runs over the set of all proper subgroups of K with the convention that Thus, for Ir' = 1, V ( K )= V / J ( R ) V . It should be pointed out that in most papers on the subject 1 n v ( V ~ is ) denoted by V H ;we shall use this notation only in the case where V = A
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is a G-graded algebra so as not to conflict with our notation for induced modules. Following Brou6 (1985), we refer to the natural surjection
BTL : Inv(Vh.) --+ V ( K ) as the Brauer morphism . We now proceed to record some basic properties of the Brauer morphism.
Lemma 3.1. Let H K be subgroups of G. Then (i) Inv(l/h) is un R(NG(li)/K)-moduZe.
(ii)
( C H C K Vi)+J(R)Inv(Vh)is an R(NG(K)/li)-submod2ze o f ~ n v ( v X ) .
(iii) V ( K ) is an R(NG(K)/K)-module annihilated by J ( R ) . Proof. It is clear that (iii) is a consequence of (ii) and the definition of V ( K ) .If g E NG(K) and v E l n v ( V ~ )then , for any z E K, z(gv) = g(g-lzg)v = gv
proving that I n v ( v ~ is ) an R(NG(li))-module. Because li’ acts trivially on Inv(Vh),it follows that Inv(Vh-) is an R(NG(K)/li)-module in a natural way. This proves (i). We are left to verify (ii). If K = 1, there is nothing to prove. Suppose that H is a proper subgroup of K and let T be a left transversal for H in Ii. Given v E 1 n v ( V ~ )and g E N ~ ( l i >we, then have
= TrgHg-I(gv) K E
vSg-,
as required.
Corollary 3.2.
For any subgroup K of G, the Brauer morphism BrK : ~ n v ( ~ + h )V ( K )
is a homomorphism of R( NG(K)/K)-VZOdUkS Proof. This is a direct consequence of Lemma 3.1. W
3 The Brauer morphism
783
Next we investigate the behaviour of the R(NG(I<)/K)-module V ( K ) .
Lemma 3.3. Let Ii' be a subgroup of G. Then (i) If V = U @ W for some RG-modules U and W , then V(1i) E U ( K )@ W ( K ) as R(NG(K)/K) - modules
(ii) v ( I { ) = vN,(,)(K). (iii) If K acts trivially on V and K is a p-group, then V ( K )= V/J(R)V Proof. (i) For any subgroup H of h', we have
I n v ( v ~=) l n v ( U ~@) Inv(WH) Taking into account that
Tr$(Inv(UH) @ 1 n v ( W ~ )=) TrE(Inv(UH))@ Tr;(lnv(WH)) the desired assertion follows. (ii) This is a direct consequence of the definition of V ( K ) . (iii) Assume that K acts trivially on V and Ii is a p-group. If 11' = 1, then by convention V ( K )= V / J ( R ) V . Hence we may assume that Ii # 1. Then, for any proper subgroup H of K , we have
V[
c J(R)Inv(V,)
since char(R/J(R))= p. Taking into account that I n v ( r / i )= V , we therefore deduce that V ( K )= V / J ( R ) V as desired. Next we examine how the Brauer morphism behaves with respect to relative trace maps
Lemma 3.4. For any subgroup K of G and any RG-module V , we
Proof. (i) The first equality is a consequence of the fact that li acts trivially on V ( K ) .Thus we need only show that for all v E Inv(VK),
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784
Let T be a full set of double coset representatives for ( N c ( K ) , K ) in G containing 1. Then, by Lemma 18.8.6(i) in Vol.1,
Kext we fix t E T - {l},put L = tKt-' n Nc(li) and denote by S a full set of double coset representatives for ( K ,L ) in N c ( K ) . Then each SLS-' n K is a proper subgroup of K and so, by Lemma 18.8.6(ii) in Vol.1,
TrP(")(tv) E
c
C T rfis-l nK( I n v (VsLs-inK)) SES
Iier(BrK)
This establishes (l),by applying (2). (ii) This is a direct consequence of (i). Assume that U , V and W are RG-modules. An R-bilinear map
f:UxV+W is said t o be G-stable if
f ( g u , g v ) = g f ( u ,w ) for all g E G, u E U , v E V The following lemma provides some properties of G-stable bilinear maps. Lemma 3.5. Let U V and W be RG-modules and let f : U x V be a G-stable R-bilinear map, For each u E U , v E V , define
by the rule :
Then the maps
are RG-homomorphisms.
---f
W
3 The Brauer morphism
u E
785
Proof. It is plain that both maps are R-linear. Kow fix g E G and U . Then, for all u E V ,
proving that f g u = g f u . Thus u +, f u is an RG-homomorphism. A similar argument shows that fg,, = g f v for all g E G , u E V . Thus u I+ f v is also an RG-homomorphism. The main property of G-stable R-bilinear maps is given by the following result.
Theorem 3.6. Let U , V and W be RG-modules and let f:UxV-+W
-
be a G-stable R-bilinear map. Then, f o r any subgroup I< of G, the map fK : U ( K )x
V(K)
W(Ii)
defined by fh.(BrK(u),Brj&)) = BTF( f ( U , Q ) ) 1 n u ( U ~ )u, E 1 n v ( V ~ )is) a well-defined ( N ~ ( l < ) / I i ) - s t a bR-bilinear le map.
(u E
Proof. Given u E 1 n v ( U ~and ) TJ E Inu(VK), it follows that for all g E K, s f ( u , 4 = f b , 9 4 = f ( u ,4 which shows that f ( u ,u) E Inv(W,v). To prove that fK is well-defined, it suffices to verify that if u E K e r ( B r K ) or u E K e r ( B r K ) ,then B r F ( f (21, u)) = 0 Assume that u E K e r ( B r g ) . Then
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for some X E J(R), x E I n v ( U k ) , U H E U z , where H runs over the set of proper subgroups of ' A (if K = 1, then put all U H = 0). Thus H
and therefore it suffices to show that B r E ( f ( U H , v)) = 0 for all H C K. Now U H = &T t y for some y E l n v ( U ~ )where , T is a left transversal for H in I<. Because v E Inv(VK), y = l n v ( U ~and ) f is G-stable, we have f(y,v) E I ~ ~ ( W H Hence ).
f ( U H v) 7
= Cf(ty,v) 1€T
= Ctf(y,v) t€T
= T r 3 f (y, 4) E Ker(BrF)
A similar argument shows that if v E l < e r ( B r i ) , then B r z ( f ( u , v ) )= 0 and thus f is well-defined. It is plain that f is an R-bilinear map. Moreover, for any g E N G ( K ) , we have
fh.(sBrm,sBra4) = = = =
fK(BTK(94,
NkP))
~rF(f(9~7gz.)) sBr.Kw(f(., 4) S f K ( B 6 2 4 ,B r 2 4 )
by applying G-stability o f f together with Corollary 3.2. This completes the proof of the theorem. 1 Now we concentrate on permutation modules. Theorem 3.7. Let V be an RG-module, let P be a p-subgroup of G such that V p is a permutation RP-module with an R-basis X permuted by P , and let V = V / J ( R ) V . Denote by 2 1 , . . . , x , all distinct elements of X fixed by P . Then the following properties hold : (i) The set {Br;(x;)Il 5 i 5 n } is an F-basis of V ( P ) . (ii) The F(NG(P)/P)-modulesV ( P ) and V ( P ) are canonically isomorphic.
3 The Brauer morphism
787
(iii) If V * is the contragredient of V and the basis X * = { X * ~ X E X } of V * is dual to X , then there is an F(NG(P)/P)-isomorphism V*(P)
---f
V(P)*
which sends the basis {Br$*(z*)ll5 i 5 n } of V * ( P )onto the dual basis of { ~ r ; ( z L . ; )5 [ l i 5 n}. Proof. (i) Let q , z 2 , . . . ,z n , %,+I,, . . ,z t be all representatives for the P-orbits of X , and let Q; be the stabilizer of 2; in P , 1 5 i 5 t. Due t o our choice of xi, we have = Q2 =
Q1
* * a
= Qn = P
and each Qn+l,. . . ,Qt is a proper subgroup of P . Owing to Lemma ll.l.l(iii),
{Tr&)ll
I: i 5 t }
is a n R-basis for Inv(Vp). Thus we need only verify that for any proper subgroup Q of P , VQ' is contained in M , where hl is the sum of J ( R ) l n v ( V p ) and the R-linear span of T r g b ( z k ) ,n 1 5 k 5 t. To this end, fix z E X and denote by H and L the stabilizers of x in Q and P,respectively. Because the elements v E V of the form v = T rQH ( z ) form an R-basis for l n v ( V Q ) ,we need only verify that for any such v, we have T r g ( v ) E M . Taking into account that
+
TTQP(TTS(.))
= Trf;(z)
=
TTI(TTfj(Z))
and T r f i ( z )E J ( l l ) I n v ( V ~whenever ) L # H , the required assertion follows. (ii) We first observe that V p is a permutation FP-module with F-basis
x = (312 E X } where 3 = z
+ J ( R ) V . Therefore, by (i), {B~:(Z;)~I 5 i 5 n }
is an F-basis of V ( P ) . Thus
V(Y) BrpV(z;)
+ H
V(p)
BrF(3;)
p-Permutation Modules
788
is an F-isomorphism. Because this isomorphism clearly preserves the action of N c ( P ) / P ,the required assertion is established. (iii) Let a, b E X and let ga = b for g E P . Then ga*(b) = a*(g-lb) = a*(a) = 1 and, for
P
# b, P
EX,
ga*(x) = a*(g-lx) = o
Consequently, ga* = b* and similarly if ga* = b*, then ga = b. In particular, it follows that x;, . . . ,x; are all distinct fixed points of X * in P. Thus, by (i), {Br;*(xf)ll 5 i 5 n} is an F-basis of V * ( P ) . We deduece therefore that the map
{
V * ( P ) 4 V(P)* B T ; * ( ~ ? )H (BrpV(x;))*
is at least an F-isomorphism. Now let f : V x V* R be defined by f(v,cp) = ~ ( v for ) all v E V , y E V'. Then, regarding R as the trivial RG-module, f becomes a G-stable R-bilinear map. Because R ( P ) = R/J(R) = F , it follows from Theorem 3.6 that the map --f
x V * ( P )-+ F
fp : V ( P )
defined by
f P ( B r 2 4 ,Brg*(cp))= cp(v>t J(R) (v E Inw(Vp),cp E Inw(V,')) is an (Nc(P)/P)-stable F-bilinear map. Thus, by Lemma 3.5, the map
{
V*(P) B.pV*(cp>
-+
V(P)* fv
where fv(Br;(w)) = ~ ( vt) J ( R ) ,is an F(Nc(P)/P)-homomorphism. Setting y = xr, we see that this homomorphism coincides with $ and the result is established. We now take a close look at the module V ( I i ) ,where I i is any subgroup of G and V is an H-projective RG-module, for some subgroup H of G.
Lemma RG-module. conjugate to W G ( K )= 0
3.8. Let H be a subgroup of G and let V be an H-projective Then, for a n y subgroup I i of G, V ( K ) = 0 unless Ii is Ga subgroup of H . In particular, if W is any RH-module, then unless A' is G-conjugate to a subgroup of H .
3 The Brauer morphism
789
Proof. Because V is H-projective, it follows from Theorem 18.9.8 and Lemma 18.9.6(ii) (both in Vol.1) that
1v
E
c
TrE(EndRH(V)) T r E n g H g - l (EndR(KngHg-l)(V)) g€G
For each g E G , we may therefore choose 1~ =
& E E n d R ( K n g ~ j g - ~ )such ( V ) that
K C TrKngHg-1
($9)
SEG
and therefore
However, for any v E Inv(V-), we have
thus completing the proof.
) , ~ R His the Our next aim is to compute explicitly ( l ~ ~ ) ' ( pwhere trivial RH-module and P is ap-subgroup of G . This will be achieved with the aid of Lemma 3.8. Note that, by Lemma 3.8, we may harmlessly assume that P is G-conjugate to a subgroup of H . Theorem 3.9. Let P be a p-subgroup of G , let H be a subgroup of G such that P is G-conjugate to a subgroup of H and let T be a set of all representatives t for the double cosets NG(P)tH for which P C tHt-'. Then
as F(NG(P)/P)-modules.
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Proof. Let X 2 T be a full set of representatives for ( N G ( P ) , H ) double cosets in G. Then, by Mackey decomposition,
where V, is defined by
On the other hand, by Lemma 3.8,
Moreover, if P E z H x - ~ ,then P acts trivially on V, and so, by Lemma 3.3(iii), we have
The desired conclusion is now a consequence of (3), (4), (5) and Lemma 3.3(i), (iii). H We now apply the Brauer morphism to a very important special case where the RG-module V is some G-algebra A. First, we introduce some notational conventions. Given a G-algebra A over R and subgroups H C 'h of G, we put
where A H is the subalgebra of H-invariant elements of A and
TT; : AH --i A" is the relative trace map. Next we observe that
3 The Brauer morphism
791
It follows from (6) and (7) that A; "(AH)
is an ideal of AK. Observe also that
= ASH9-I
for all g E G
(8)
and that
Because the G-algebra A is an RG-module, we may define A ( K ) as we did for any RG-module V , i.e. by putting
A ( K ) = A"/
L
C A E + J(R)A"
where H runs over the set of all proper subgroups of that = 0 for K = 1
) K with the convention
EAi H
In particular, the natural surjection
B r i - : AK
+
A(li)
gives us the Brauer morphism with respect t o the RG-module V = A . Lemma 3.10. Let li be a subgroup of G and let A be a G-algebra over
R. Then
(i) l i e r ( B r ; ) is an ideal of A". (ii) A" is an (NG(li)/li)-algebraover R, while A ( K ) is an ( N G ( K ) / K ) algebm over R annihilated by J ( R ) . (iii) The Brauer morphism
B r i : A"
-, A ( K )
is a homomorphism of (NG(lr')/K)-algebras,i.e. a homomorphism of Ralgebras preserving the actions of N G ( K ) / K on AK and A(1i). Proof. (i) This follows directly from the fact that both A; and J(R)A" are ideals of A". (ii) The assertion concerning A" follows from (8) applied to H = K . The remaining assertion is a consequence of (9) and the definition of A ( 1 i ) . (iii) Apply (i) and Corollary 3.2. H
p-Permutation Modules
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We close by providing the following observation which justifies the term “Brauer morphism”.
Lemma 3.11. Let A = R X be the group algebra of a group X over R and let G act on X as a group of automorphisms of X . Then A becomes a G-algebm in the natural way and the following two properties hold : (i) If P is a p-subgroup of G and H is the subgroup of all elements of X fixed by P, then the map
{
CheHhBr$(h)
--f
FH
I+
ChcHAhh
(Ah E
F)
is an isomorphism of F-algebras. (ii) If R = F , X = G and G acts on itself by conjugation, then H = CG(P) and, upon the identification of FG(P) with FCG(P), the restriction of BT;’ to Z ( F G ) is the classical Brauer homomorphism (see Sec.3 of Chapter 8 in Vol.1).
Proof. (i) This follows from Lemma 3.lO(iii) and Theorem 3.7(i). (ii) Let C1,. .. ,C,, be all P-conjugacy classes of G. If ICil > 1, then BTFG(C?) = 0
while, for all h E CG(P),
BrFG(h)= h
This clearly implies the desired assertion.
4
Scott modules
In what follows, we fix a prime number p and a finite group G. We also assume that R is either a complete discrete valuation ring with c h a r ( R / J ( R ) )= p or R is a field of characteristic p , and put
F = R/J(R) All RG-modules are assumed t o be finitely generated. To emphasize the coefficient ring R, we write 1 ~ ’for the trivial RG-module.
4 Scott modules
793
Our principal goal is to examine an important type of indecomposable p-permutation RG-modules, the so called Scott modules . Such modules were first discovered by L.L. Scott (unpublished) and the first published account can be found in Burry (1982). According to Burry (1982, p.1856), Scott modules were later rediscovered by Alperin. Our method below is based entirely on a work of Brou6 (1985) who utilized a systematic use of the Brauer morphism. All the relevant background concerning Brauer morphisms can be found in the preceding section. We begin by introducing the following definition due to Puig (see Brou6 (1985)). Let V be a p-permutation RG-module and let P be a p-subgroup of G. Then the Scott coefficient of V associated with P , written sp(V) is defined by sp(V) = dimF(BrpV(V,G)) where
Br,v : Inv(Vp)
---f
V(P)
is the Brauer morphism introduced in Sec.3 and
Observe that, by Lemma 3.4(ii) applied t o K = P , we have
Our first task to investigate the behaviour of s p ( V ) . Lemma 4.1. Let P be a p-subgroup of G. (i) If V and V' are two p-permutation RG-modules, then
sp(V @ V')= S P ( V )
+
SP(V')
(ii) If V is a p-permutation RG-module and V* is the contragredient of
V , then
+(V) = SP(V*)
(iii) For any subgroup H of G, s p [ ( l ~ ~= )1~if ]P is G-conjugate to a Sylow p-subgroup of H and s p [ ( l ~ ~ ) = ' ] 0 otherwise. Proof. (i) Apply Lemma 3.3(i) and (1). (ii) Invoking Theorem 3.7(iii), we see that sp(V*) is equal t o the rank of
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the endomorphism T r NG(p)’p , of V(P)*. The latter is equal t o the rank of its transpose, i.e. of the endomorphism T r N , ~ ( p ) ’ pof v ( P ) . Consequently,
w(V) = V ( V * ) as desired. (iii) Owing t o Lemma 3.8, ( ~ R H ) ~=(0Punless ) P is G-conjugate to a subgroup of H. By Sylow theorems in H ,if g-l P g is a Sylow p-subgroup of H for some g E G,then g r l P g l is a Sylow psubgroup of H only for those g1 E G for which g1 E NG(P)gH. Invoking Theorem 3.9, it therefore suffices t o show that has dimension 1 or 0 according t o whether H is , or is not, a p’-group. By Lemma 11.1.3(ii), we may identify ( ~ F H with ) ~ the submodule FG. H+ of FG. Then we have
((1~~)~)7
(( ~
F H ) ~ )= ?
TrF(FG
*
P)
= G+.FG.H+ = FG.G+.H+ = IHI(FG+),
as desired. We can now obtain our desired goal.
Theorem 4.2. Let P be a p-subgroup of G. Then (i) There exists a unique (up to RG-isomorphism) indecomposable ppermutation RG-module Sp(G,R) such that sp(Sp(G,R ) ) # 0. Moreover, sp(Sp(G,R)) = 1, and
SP(G, R) 2 [Sp(G, R)]* (ii) If H is a subgroup of G, then Sp(G, R ) is isomorphic to a direct summand of ( 1 ~if and ) ~only if P is G-conjugate to a Sylow p-subgroup of H . In particular, by taking H = P, S p ( G ,R) has vertex P.
Proof. (i) Owing t o Theorem l.l(vi), any indecomposable p-permutation RG-module V is a direct summand of ( I R Q ) for ~ some p-subgroup Q of G. If s p ( V ) # 0, then by Lemma 4.1(i), (iii), we see that P is G-conjugate to Q . Moreover, since s p ( ( l ~ p ) = ~ )1, it follows that V is the unique indecomposable direct summand of ( 1 ~ psuch ) ~ that s p ( V ) # 0. Hence s p ( V ) = 1
4 Scott modules
795
and, by the unicity of V , it follows from Lemma 4.l(ii) that V 2 V'. (ii) It is a consequence of (i) and Lemma 4.l(i) that Sp(G, R ) is isomorphic t o a direct summand of ( ~ R H ) ' if and only if
#0
sP((1RH)')
Thus, by Lemma 4.l(iii), P is G-conjugate t o a Sylow p-subgroup of H if and only if S p ( G , R ) is isomorphic t o a direct summand of ( ~ R H ) This ~ . concludes the proof of the theorem. W We shall refer to the RG-module Sp(G,R) as the Scott module of G associated t o P . The aext result provides some further properties of Scott modules.
Let P be a p-subgroup of G and let H be n subgroup of G such that P is G-conjugate to a Sylow p-subgroup of H. Then (i) H O ~ R G ( ~SP(G, R G ,R ) ) 2 H o ~ R G ( S P ( G R ), ,~ R G 2 ) R. (ii) Sp(G, R ) is the unique indecomposable direct summand V of ( ~ R H ) ' such that H O ~ R G ( ~VR) # G 0, Theorem 4.3.
(iii) Sp(G,R ) is the unique indecomposable direct summand V o f ( 1 ~ ~ ) ' such that HomRG(V, 1 R G ) # 0 Proof.
Owing to Proposition 18.1.6 in Vol.1, we have HomRG(lRG, (lRP)G)
HOmRP(1RP, 1 R P )
S
R HomRG'(((lRP)G,lRG)
Since, by Theorem 4.2(i), S p ( G ,R ) 2 Sp(G, R)* it suffices to show that HomRG(lRG, SP(G, R ) ) # 0
(2)
To this end, we observe that, by Lemma 3.4(ii), (Sp(G,R))$ is mapped onto ( S d G ,R ) ) ( P N ) ,G ( p ) ' p ; this last module is not zero by the definition of Sp(G, R ) because its dimension is precisely sp(Sp(G, R ) ) . Thus
W S P ( G , R ) )# 0 which establishes (2). This concludes the proof of the theorem. H
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Next we relate S p ( G , R ) with Sp(G,F) and examine t o what extent S p ( G ,R ) determines P.
Theorem 4.4. Let P and Q be p-szlbgroups of G. Then (i) SP(G,R ) / J ( R ) S P ( GR, ) &(G, F ) . (ii) SQ(G,R ) S Sp(G,R ) if and only if Q is G-conjugate to P . Proof. (i) Let V be any ppermutation RG-module and let
v = V / J (R)V Owing to Theorem 3,7(ii), we have sp(V) = s p ( v ) . Thus by the characterization of Sp(G, R ) given by Theorem 4.2(i), we deduce that
Sp(G,R )
S d G ,F)
as required. (ii) This is a direct consequence of Theorem 4.2(ii). H
The next result examines two extreme cases for R = F .
Corollary 4.5. Let P be a Sylow p-subgroup of G . Then (i) S p ( G , F ) ~ F G . (ii) S I ( G , F ) is a projective cover of ~ F G . Proof. (i) Apply Lemma 11.1.3(v) and Theorem 4.3(ii). (ii) Apply Theorem 4.3(iii). H The rest of this section is based on a work of Burry (1982). In what follows, F denotes an arbitrary field of characteristic p > 0. Let us examine the Green correspondents of Scott modules.
Theorem 4.6. Let P be a p-subgroup of G , let V = Sp(G,F) and let H = N G ( P ) . Then the Green correspondent f ( V ) of V with respect to ( G ,P , H ) can be considered as an F(Ei/P)-module and as such f ( V ) is a projective cover of l F ( H / p ) . Proof. By Theorem 4.2(ii), P is a vertex of V . Hence f ( V ) is indeed defined. Since the Green correspondence preserves vertices and sources, we
4 Scott modules
797
see that P is a vertex of f ( V ) and 1 ~ ispa source of f ( V ) . Thus f ( V ) is a component of ( 1 ~ p ) But, ~ ~ since . P a H , we have ( 1 p p ) H 2! F ( H / P )
by virtue of Theorem 18.5.1 in Vol.1. Thus f ( V ) can be regarded as an F(H/P)-module and as such f ( V ) is a projective indecomposable module. On the other hand, since
0
#
HOmFG(lFG,V)
C HomFG(lFG, f(V)')
(by Theorem 4.3(i)) (by Theorem 7.2.1)
HomFH(lFH,f(V)) (by Proposition 18.1.6(ii) in Vol.1) it follows that the FH-module f (V) has a trivial submodule. Hence the same is true for the F(H/N)-module f(V). This shows that Soc( f(V)) 2 l p ( ~ / q and hence f(V) is a projective cover of l ~ ( ~ / pH) . Assume that H is a subgroup of G and P is a p-subgroup of H . Then P determines two Scott modules, namely
It is therefore appropriate to investigate whether the above modules are related t o each other. The next result provides some pertinent information.
Theorem 4.7. Let H be a subgroup of G and let P be a p-subgroup of H. Then the following properties hold : (i) Sp(G,F) is a component of S P ( H , F ) ~ . (ii) Sp(H, F ) is a component of [Sp(G,F ) ] H .
Proof. (i) Put Q = N H ( P ) / Y , L = N c ( P ) / P , U = S p ( G , F ) and V = S p ( H , F ) . Let X be the Green correspondent of V with respect to (H,P,N,(P)) and let Y be the Green correspondent of U with respect to
(G, P, NG(P)).
Owing t o Theorem 4.6, X can be regarded as an FQ-module and as such X is a projective cover of ~ F QSimilarly, . Y can be regarded as an FL-module
p-Permutation Modules
798
and as such Y is a projective cover of ~ F L . Now there is a surjective F Q homomorphism X + ~ F and Q so there is a surjective FL-homomorphism of induced modules : X L (1FQY On the other hand, since
H o ~ F L ( ( ~ F Q ) ~2, ~ HFOL~)F Q ( ~~FFQQ,#) 0 there is a surjective FL-homomorphism (lFQ)L
+
1FL
Thus there is a surjective FL-homomorphism
xL + 1 F L Since X L is a projective FL-module and Y is a projective cover of ~ F L it, follows from Proposition 7.1.2 in Vol.1 that Y is a component of X L . Hence the FNG(P)-module Y is a component of the F(Nc(P))-module XNG('). This proves (i), by applying Theorem 7.2.5(ii). (ii) There is a surjective FL-homomorphism Y -+ ~ F and L so there is a surjective FQ-homomorphism YQ + ~ F Q .Since YQ is a projective F Q module, i t follows from the argument in (i) that the F(Nff(P))-module X is a component of Y N H ( p ) .This proves (ii), by applying Theorem 7.2.5(i). H Note that Theorem 4.7(i) tells us that if we induce a Scott module with vertex P , we get a Scott module with vertex P as a component. The next theorem is a converse of this. More precisely, we have the following result. Theorem 4.8. Let H be a subgroup of G and let P be a p-subgroup of H . If U is an indecomposable p-permutation RH-module and Sp(G, R ) is a component of U G , then U 2 Sgpg-l( H ,R ) , for some g E G with gPg-' H.
Proof. By Theorem 1.1, U is a component of ( ~ R L for ) some ~ subgroup L of H . Setting S = Sp(G,R ) , it follows that S is a component of ( ~ R L ) ~ . Hence, by Theorem 4.2(ii), P is G-conjugate to a Sylow p-subgroup, say Q, of L. Moreover, since 0
#
c 2
HOmRG(lRG,S) HomRG( 1RG, uG)
H O ~ R H~ R( H u , )
5 Vertices of p-permutation modules
799
it follows from Theorem 4.3(ii) (with W = L , G = H and P = Q ) that U S S Q ( H , R ) . This completes the proof of the theorem. W
5
Vertices of p-permutation modules
We preserve all notation and conventions introduced in the previous section. Our aim is t o investigate vertices of p-permutation modules. This will be achieved by applying a number of properties of the Brauer morphism t o ppermutation modules. Our point of departure is the following two auxiliary assertions, both extracted from a work of Broud (1985).
Lemma 5.1. Let P be a p-subgroup of G and let V be a p-permutation RG-module. Then V ( P ) is a p-permutation F(NG(P)/f')-module. Proof. Denote by Q a Sylow p-subgroup of N G ( P ) .Then, by Theorem 1.1, VQ is a permutation RQ-module. Choose an R-basis X of V which is permuted by Q. Then the F-basis for V ( P ) given by Theorem 3.7(i) is permuted by Q / P , as asserted. H
Lemma 5.2. Let P be a p-subgroup of G , let V be a p-permutation RG-module, and let A = EndR(V). Then the map
where
is an isomorphism of F(Nc(P)/P)-algebras. Proof. The natural bilinear map f : A x V G-stable. Thus, by Theorem 3.6, the map fp : A ( P )x
V ( P )--+ V ( P )
--+
V , (a,v) H a(.)
is
p-Permutation Modules
800
is a well-defined (NG(P)/P)-Stable F-bilinear map. Therefore, by Lemma 3.5, the given map is a homomorphism of F(NG(P)/P)-algebraS. Let X denote an R-basis of V which is permuted by P. Then the set
where ax,y E A is defined by
al,y(z) = Sy,*x
for all
2,
y,z E X
is an R-basis of A which is permuted by 1'. Let C x ( P )be the set of fixed points of X under P . Then we clmrly have cX(A)(P)
= {ax,ylx, Y E c X ( P ) }
Given z E Cx(P)and a E C X ( A ) ( P )we , now put Brj!(a). Then we must have
a:
= B r F ( x ) and u =
It therefore follows from Theorem 3.7(i) that the given homomorphism is an isomorphism. H We are now ready to investigate vertices of p-permutation modules.
Theorem 5.3. (Broue' (1985)). Assume that V is a n indecomposable p-permutation RG-module. Then (i) The vertices of V are the maximal p-subgroups P of G such that V ( P ># 0. (ii) A p-subgroup P of G is a vertex of V if and only if V ( P ) # 0 and V ( P ) is a projective F(NG(P)/P)-module. Proof. (i) Assume that a p-subgroup P of G is a vertex of V . Then, by Lemma 3.8, it suffices to verify that V ( P ) # 0. We know that V is a component of ( 1 ~ p ) Hence, ~ . by Lemma 11.1.9, Vp is isomorphic to a direct sum of modules of type : [lR(PrlgPg-')I
P
where g runs over a certain subset S of G. But V is a component of and because P is a vertex of V it follows that S n NG(P) # 0. Thus 1 ~ isp a component of V p and so V ( P )# 0.
5 Vertices of ppermiitatiori modules
801
(ii) Suppose that V ( P ) # 0 and that V ( P ) is a projective F ( N G ( P > / P j module. Setting A = E n d R ( V ) , it follows from Lemma 5.2 and Theorem 18.9.8 in Vol.1 that A(p)NC(")Ip = ( A ( P ) ) F ( p ) / p But, by Lemma 3.4(ii), ( A ( P ) ) ,N G ( p ) ' p is the image of A: under the Brauer morphism Br;?. Because l v is the unique nonzero idempotent of AG, we deduce that l v E A:, proving that V is P-projective. Hence, by (i), P is a vertex of V . Conversely, suppose that P is a vertex of V . Then l v E A: and therefore
Hence, by Lemma 5.2, V ( P ) is a projective F(NG(P)/P)-module. H Next we enumerate all indecomposable p-permutation modules with vertex P via the isomorphism classes of projective indecomposable F ( N G ( P ) / P ) modules.
Theorem 5.4. (Broue' (1985) and Klyachko (1979)). Let V be an indecomposable p-permutation RG-module and let P be a p-subgroup of G. Then the map V H V ( P ) induces a bijective correspondence between the isomorphism classes of indecomposable p-permutation RG-modules with vertex P and the isomorphism classes of projective indecomposable F ( NG(P)/P)modules. In particular, S d G ,W ( P )
is the projective cover of lF(NG(P)IP)
Proof. Owing to Theorem 3.9, we have ( ( l R P ) G > ( P ) (IF) N o ( P ) I P
The indecomposable p-permutation RG-modules with vertex P correspond to the indecomposable components of ( 1 ~ p with ) ~ vertex P. Observe also that the projective indecomposable F(NG(P)/p)-modules correspond to indecomposable components of
p-Permutation Modules
802
Now put A = EndR((lRp)G). Then, by Lemma 5.2,
as F(NG(P)/P)-dgebras. The indecomposable components of ( l R p ) G with vertex P correspond to the primitive idempotents of AF whose image in A(P) is nonzero. Because, by Lemma 3.4(ii), B r i maps A$ onto A(P),N G ( P ) / P
and AG = A;, we have
Thus the first assertion follows from the standard facts about lifting idempotents : an indecomposable component V of (lRp)G with vertex P corresponds to a primitive idempotent e of A: such that Brj!(e) # 0, which corresponds to the primitive idempotent B r $ ( e ) of ( A (P ) ) N G ( P ) / P , which in turn corresponds to the direct summand
of ( ~ F ) ~ G ( ' ) / ~The . assertion regarding Sp(G,R)follows from the fact that, by definition, ( S p ( G ,R)(P))NG(P)/P # 0 (alternatively, it follows from the equality S p ( G ,R ) ( P )= Sl(NG(P)/P,F ) .
Corollary 5.5. Suppose that P is a p-subgroup of G . Then the number of nonisomorphic indecomposable p-permutation RG-modules with vertex P is equal to the number of F-conjugacy classes of p-regular elements of
/ p.
NG ( P)
Proof. Apply Theorem 5.4 together with the Witt-Berman's theorem (Theorem 17.5.3 in Vol.1). The following consequence of Theorem 5.4 provides an alternative proof of Theorems 1.4(i) and 1.5(i). Corollary 5.8. The reduction modulo J ( R ) provides a bijection between the isomorphism classes of p-permutation RG-modules and the isomorphism classes of p-permutation FG-modules.
5 Vertices of p-permutation modules
803
Proof. Owing to Theorem 3.7(ii), V ( P )2 v ( P ) where V and P are as in Theorem 3.7. The desired assertion is therefore a consequence of Theorem 5.4. H Another interesting consequence of Theorem 5.4 is given by
Corollary 5.7. (Bmue' (1985)). There is a bijection between the isomorphism classes of indecomposable p-permutation RG-modules and the Gconjugacy classes of pairs ( P , E ) , where P is a p-subgroup of G and E is a projective indecomposable F( NG( P)/P)-module. Proof. This is a direct consequence of Theorem 5.4. H We now probe more deeply into the nature of the correspondence V V ( P )given by Theorem 5.4.
I+
Theorem 5.8. (Bmue' (1985)). Let P be a p-subgroup of G , let M be Q p-permutation RG-module and let E be a projective indecomposable F ( N G (P ) / P)-module. If N is the corresponding indecomposable p-permutation RG-module with vertex P , then N is a component of M if and only i f E is a component of M ( P ) .
Proof. Note that, by definition, the RG-module N is the indecomposable p-permutation RG-module determined by the condition N ( P ) = E . It is plain that if N is a component of M , then E is a component of M ( P ) . Conversely, assume that E is a component of M ( P ) , and put A = E n d ~ ( A 4 )Then, . by Lemma 5.2, A ( P ) S EndF( M ( P ) )
contains a primitive idempotent e such that e M ( P ) E E . Using the standard facts about lifting idempotents together with Lemma 3.4, it follows that A: contains a primitive idempotent f such that B r $ ( f ) = e. Thus f - M is an indecomposable component of M with vertex P , and (f - M ) ( P ) = e . M ( P ) 2 E . This implies that f . M 2 N and so N is a component of M , as desired. H
p-Permutation Modules
804
The next result demonstrates that V ( P ) is a familiar classical object for the case where V is an indecomposable p-permutation FG-module with vertex P.
Proposition 5.9. (Broue‘ (1985)). Let V be a n indecomposable ppermutation FG-module with vertex P . Then the Green correspondent of V with respect to (G,P,Nc(P)) is the FNc(P)-module V(P). Proof. Let W be the Green correspondent of the FG-module V with respect t o (G, P, N c ( P ) ) . Since, by Theorem 1.1, V has a trivial source, W must also have a trivial source. Thus W is isomorphic to a direct summand of (lFP)NG(P)
and so P acts trivially on W , which forces W ( P )= W . But, by Theorem 7.2.1, VNG(P)2 W 63 W’ where W’ is a direct sum of indecomposable F(Nc(P))-modules with vertex strictly contained in P. It therefore follows from Lemma 3.8 that W ’ ( P )= 0. Thus we must have
E
W(P)
(by Lemma 3.3(i))
= w as desired. H Some further interesting properties of p-permutation modules can be found in a work of Klyachko (1979).
6
Puig’s theorem
Throughout, G denotes a finite group and F an arbitrary field of prime characteristic p. All FG-modules are assumed to be finitely generated. Given a p-permutation FG-module V , we may form two NG(P)-algebras
6 Puig’s theorem
805
(all relevant information and notation is contained in Sec.3). The importance of the FNc(P)-module V ( P )stems from the fact that if V is indecomposable with vertex P , then V ( P )is the Green correspondent of V with respect to (G, P, NG(P)) (see Proposition 5.9). We remind the reader that, by Lemma 5.1, V ( P )is also a p-permutation F(Nc(P)/P)-module. Our principal god is to prove that the NG(P)-algebras in (1) are identifiable. This will be achieved with the aid of the following result. Proposition 6.1. (Cabanes (1988)). Let P be a normal p-subgroup of G and let V be a p-permutation FG-module. Let U be the direct sum of those indecomposable direct summands of V whose vertex contains P and let W be the sum of the others. Then (i) V = U @ W , V ( P )2 U and W ( P )= 0 . (ii) I n v ( U p ) = U and I n v ( W p ) = Ker(Br:). (iii) For A = E n d F ( V ) , we have
Proof. (i) and (ii). It is plain that V = U @ W .Moreover, by Theorem 5.3(i), W ( P ) = 0 and, by the definition of U , Inv(Up) = U . Because I n u ( U p ) = U , we have K e r ( B $ ) = 0 (hence U ( P ) = U ) , while because W ( P )= 0 we have K e r ( B r y ) = I n v ( W p ) . Consequently,
V(P)
U(P)$W(P) 2 U(P)=U
and
which proves (i) and (ii). (iii) Let Q be a proper subgroup of Y and let f E EndFQ(V).To prove “E” containment, it suffices to verify that
TrQP(f)(u)E TV Because u E Inv(Vp), we have
for all
uE
u
p-Permutation Modules
806
with f(u) E I ~ V ( V Q But ) . then
TrC(f(4) E v,p g Iier(BrpV) = I n v ( Wp ) as required. To establish the reverse containment, let e : V -+ U be the projection map, and let 1- e = Cy.l e; be the decomposition of 1- e associated with a decomposition of W as a direct sum of indecomposable FG-modules whose vertices do not contain P. Let Pj be a vertex of ej(W),1 5 i 5 n. Then P; does not contain a G-conjugate of P and ej E A:,, 1 5 i 5 n. However, if i~ {1,..., n } , then
gPjg-' IIP
cP
for all g E G
which forces ei E A:,
G
c
C~rp*~-Inp g€G
Iier(Br$)
and therefore 1 - e E Ker(Br$). Thus, i f f E E n d ~ p ( Vwith ) f(U)C W, then ef e = 0 and
f as desired.
+
= (1 - e ) f e f ( 1 - e ) E Ker(Br$)
m
We are now ready to achieve our goal. The following theorem of Puig can be found in a work of Cabanes (1988).
Theorem 6.2. Let V be a p-permutation FG-module and let P be a p-subgroup of G. Then
( E n d F ( V ) ) ( PE ) EndF(V(P))
US
N G ( P )- algebras
Proof. We first observe that Inv(Vp) and K e r ( B r F ) are stable under every element of E n d ~ p ( V ) .Hence the natural action of E n d ~ p ( Von ) V ( P )induces a homomorphism II,: EndFp(V)+ E n d ~ ( v ( P ) )
6 Puig’s theorem
807
of Nc(P)-algebras. It will next be shown that this homomorphism is surjective and has the desired kernel. Write VNG(p)= U @ W , where U is the direct sum of those indecomposable direct summands of V ~ ~ ( whose p 1 vertex contains P and where W is the sum of the others. Owing t o Lemma 3.3(ii),
‘(1
= ‘NG(p)(‘)
Therefore, by Proposition 6.1(i), V ( P )2 U . It is plain that the homomorphism .Ic, corresponds to the map
En&p(V)
f
--$
EndF(U) efe
where e : VNG(p) --f U is the projection map, by identifying E n d ~ ( uwith )
It therefore follows that
Thus, by Proposition 6.l(iii), KeT.Ic, = K e r ( B $ ) for A = E n d ~ ( v ) . By the foregoing, we are left to verify that .Ic, is surjective. By Proposition 6.1(ii), we have U = I n v ( U p ) and so E n d ~ ( u = ) End,rp(U). Now fix X E E n d ~ ( uand ) denote by A‘ an element of E n d ~ p ( Vby ) extending X to V with 0 on W . Then it is obvious that .Ic, maps A‘ to X and the result is established. H
p-Permutation Modules In this chapter, we provide a thourough investigation of p-permutation modules. The coefficient ring R is chosen to be either a field F of prime characteristic p or a complete discrete valuation ring such that c h a r ( R / J ( R ) )= p. Although a given RG-module V need not be a permutation module, it is quite possible that the restriction Vp of V to RP, where P is a Sylow psubgroup of G, is a permutation module. Such an RG-module V is called a ppermutation module. It turns out that the p-permutation modules are precisely direct summands of permutation modules. The chapter is divided into six sections. In Sec.1, we introduce relevant definitions and provide various characterizations of p-permutation modules. In particular, we show that an indecomposable RG-module is a ppermutation module if and only if it has a trivial source. We then demonstrate that any ppermutation FG-module lifts uniquely to a p-permutation RG-module, and show that the corresponding residue class map is surjective. In Sec.2, we provide further characterizations of p-permutation modules via the notions of monomial and virtually monomial modules. The corresponding results are extracted from a work of Dress (1975). The section also contains some additional results of independent interest. In Sec.3, we introduce an important tool for the study of p-permutation modules, namely the Brauer morphism. Most of the results presented can be found in Brouk (1985) and Brouk and Puig (1980). The corresponding theory is then applied in Sec.4 for the study of Scott modules. In Sec.5, we provide a detailed investigation of vertices of p-permutation modules. Many properties of this nature were investigated in a work of Klyachko (1979) and Brouk (1985). One of these properties asserts that, 771
772
p-Permutation Modules
for any psubgroup P of G , the number of nonisomorphic indecomposable p-permutation RG-modules with vertex P is equal to the number of Fconjugacy classes of pregular elements of N G ( P ) / P . In the final section, Sec.6, we prove an interesting result due to Puig which asserts that the NG(P)-algebras ( E n d F ( V ) ) ( P )and E n d F ( V ( P ) )are identifiable. Here V is a p-permutation FG-module and P is a p-subgroup of G.
1
Definitions and basic properties
In this section, G denotes a finite group and R is either a field of characteristic p > 0 or a complete discrete valuation ring such that c h a r ( R / J ( R ) )= p > 0. In what follows, we put F = R/J(R). If H is a subgroup of G, then all RHmodules are assumed to be finitely generated. We say that an RG-module V is a p-permutation m o d u l e if the restriction V p of V to a Sylow psubgroup P of G is a permutation RP-module. Thus a nonzero RG-module V is a p-permutation module if and only if V is R-free of finite rank with an R-basis on which a Sylow p-subgroup of G acts as a permutation group. Our point of departure is the following basic result which can be found in Dress (1975).
T h e o r e m 1.1. Let V = V1 @ ... @ V, be a direct decomposition of a nonzero RG-module V into indecomposable submodules. Then the followang conditions are equivalent : (i) V is a p-permutation module. (ii) For any p-subgroup Q of G , VQ is a permutation RQ-module. (iii) V is a direct summand of a permutation module. (iv) There exist subgroups H I , . . . ,H , of G such that, for a n y i E { 1, . . . ,n } , V, is isomorphic to a direct summand of ( 1 ~ ~ ) ~ . (v) Each V , is a p-permutation module. (vi) Each V , has trivial source. Proof. (i) 3 (ii) : Let P be a Sylow p-subgroup of G such that Vp is a permutation RP-module. Then Q C gPg-' for some g E G. If vl,. . . ,Vk is a permutation basis for V p , then g q , . . , ,gvk is a permutation basis for Vgpg-l and hence for VQ,as required. (ii) + (iii) : Let P denote a Sylow p-subgroup of G. By assumption, V p
1 Definitions and basic properties
773
is a permutation RP-module. Hence (Vp)' is a permutation RG-module. But V is P-projective, so V is isomorphic t o a direct summand of ( V p ) G , proving (iii). (iii) + (iv) : By hypothesis, there is a permutation RG-module M such t h a t each V, is a direct summand of M . Because V, is indecomposable, it follows from the unique decomposition property that V, is isomorphic to a direct summand of ( 1 ~ ~for ) ' some subgroup Hi of G, 1 5 i 5 n , proving property (iv). (iv) (v) : Because ( 1 ~ ~ is ) ' a permutation module, it is also a ppermutation module. Hence we need only show that a nonzero direct summand N o€a ppermutation module M is a ppermutation module. If P is a Sylow psubgroup of G such that Mp is a permutation RPmodule, then N p is a direct summand of M p . But Mp is a direct sum of modules of the form ( l ~ ) ' ,where Q is a p-subgroup of G. By Lemma 11.1.4, ( 1 ~ ) 'is indecomposable. Hence, by the unique decomposition property, N p is a direct sum of modules of the form ( 1 ~ )Thus ~ . N is a p-permutation module. (v) + (vi) : Put M = V, and let P be a Sylow p-subgroup of G such that Mp is a permutation RP-module. If Q P is a vertex of M , then M is Q-projective and so M is isomorphic t o a direct summand of (MQ)'. By hypothesis, MQ is a direct sum of modules of the form (ls)Q,where 5' is a subgroup of Q. Hence M is a direct summand of (1s)Q for some subgroup S of Q. Because M is S-projective and S Q , S is a vertex of M and hence M has trivial source. (vi) + (i) : It suffices t o verify that each V, is a ppermutation module. By the implication (i) (iii), V, is a direct summand of a permutation (hence p-permutation) module. Hence, by the proof of the implication (iv) + (v), V, is a p-permutation module. H
*
*
As an easy consequence, we now derive
Corollary 1.2. Let H be a subgroup o f G and let V and W be RGmodules. (i) If V = VI @ . . @ V, for some nonzero RG-modules V,, then V is a p-permutation module if and only if each I$ is a p-permutation module. (ii) If V and W are p-permutation modules, then so is V @R W . (iii) If V is a p-permutation module, then VH is a p-permutation module. (iv) If U is a p-permutation module RH-module, fhen UG is a p-permutation
p-Permutation Modules
774
RG-module. (v) If V is a p-permutation module, then so is V'.
Proof. (i) Direct consequence of Theorem 1.1. (ii) If {v;} and {wj} are R-bases of V and W , respectively, which are permuted by a Sylow psubgroup of G, then the same is true for the R-basis {v; 8 wj} of V 8~W . (iii) If Q is a Sylow psubgroup of H , then VQ= (VH)Q is a permutation RQ-module, by virtue of Theorem l.l(ii). (iv) This is obvious. (v) Let P be a Sylow psubgroup of G. Then Vp is a permutation RPmodule. Since (Vp)* Z ( V * ) p , it follows from Lemma 11.1.6 that (V*)pis a permutation RP-module. Thus V* is a ppermutation module. W Let U , V be two RG-lattices. Then, for any f E HomRG(U,V),
Hence f induces an FG-homomorphism
given by
-
Here 21 is the image of u in 0 = U/J(R)U and f ( u ) is the image of f ( u ) in V = V / J ( R ) V .We remind the reader that the additive homomorphism
is called the residue class map . Owing to Theorem 2.7.11, the residue class map induces an injective F-linear map
v)
, lift The image of this map consists precisely of all X E H o ~ F G ( ~ which to H o ~ R G ( UV, ) . We now apply the above information to the special case where U and V are permutation RG-modules.
1 Definitions and basic properties
775
Lemma 1.3. Let U and V be permutation RG-modules.
Then the
residue class map induces an F-isomorphism
If U = V , then the above is an isomorphism of F-algebras. Proof. Since a permutation module is a direct sum of transitive permutation modules, we may assume that both U and V are transitive permutation modules. Then U 2 ( l n ~ ) 'and V 2 (1~s)'for some subgroups H and S of G. Hence 2 ( ~ F H and ) ~ =" ( 1 ~ s ) It ~ .follows from Lemma 11.1.8 that both sides in (1) have the same F-dimension (which is equal t o the number of (S,H)-double cosets in G). This clearly implies the first
v
assertion. The second assertion being obvious, the result follows.
We are now ready t o record the following important property of ppermutation modules.
Theorem 1.4. The following properties hold : (i) Any p-permutation FG-module lifts uniquely (up to isomorphism) to a p-permutation RG-module. (ii) For any p-permutation RG-modules U and V , the residue class map induces an F-isomorphism
If U = V , then the above is an isomorphism of F-algebras.
Proof. (i) Let M be a p-permutation FG-module. Then, by Theorem l.l(iii), M is a direct summand of a permutation FG-module, say N . Let E E E ~ ~ FN G ) be ( the corresponding idempotent, and choose any permutation RG-module U with U = N . Owing t o Lemma 1.3 and Theorem 1.6.20 in Vol.1, E can be lifted t o an idempotent, say e , of EndRG(U). Setting V = e ( U ) it follows that V is a direct summand of U (hence V is a ppermutation module) such that
v 2 E(N) = M Thus M lifts t o a p-permutation RG-module. Suppose that Ul and V, are p-permutation RG-modules with UI 2 V I .
p-Permutation Modules
776
We may assume that U1 and V1 are direct summands of the same permutation RG-module U . Choose idempotents e , f of EndRG(U) with e ( U ) = U1 and f ( U ) = V1. Since 01 2 we have e( U ) 2 $( U ) . Hence, by Proposition 1.9.4(iii) in Vol.1, EndFG( U ) Z EndFG( U)f
q,
Therefore, by Lemma l.S(with V = U ) and Theorem 1.6.20 in Vol.1,
EndRG( u ) e !X EndRG( u )f Again, by Proposition 1.9.4(iii) in Vol.1, U1 2 Vl and (i) is established. (ii) Since a ppermutation module is a direct summand of a permutation module, the required assertion is a consequence of Lemma 1.3. As an easy application of the above, we next record the following result.
Theorem 1.5. Let U , V be p-permutation RG-modules. Then (i) U E V if and only i f 2 (ii) U is indecomposable i f and only if U is indecomposable. (iii) If U is indecomposable with vertex Q , then 0 is indecomposable with vertex Q .
u v.
Proof. (i) This is a direct consequence of Theorem 1.4(i). (ii) It is clear that if U is indecomposable, then so is U . Conversely, assume that U is indecomposable. If U = X @ Y for some nonzero FGmodules X and Y , then X 2 and Y = U 2 for some p-permutation RG-modules U1 and U2 (see Theorem 1.4(i)). Then U E UI @ U2 and, by (i) U 2 U1 @ U2. This is a contradiction, since U1 # 0 and U2 # 0. Thus U is indecomposable and the required assertion follows. (iii) Let P be a vertex of U . We must show that P and Q are G-conjugate. Since U is a component of ( U Q ) ~it,follows that U is a component of ( 0 ~ ) ~ . Hence P is G-conjugate t o a subgroup of Q. On the other hand, there exists f E E n d F p ( u ) such that 10 = T r s ( f ) . By Theorem 1.4(ii), we may write f = 1for some X E E n d R p ( U ) . Thus
ul
lu which forces
E T$(X)
(modJ(R)EndRG(U))
2 Dress’s theorems
777
This shows that U is P-projective and so Q is G-conjugate t o a subgroup of P . Thus P and Q are G-conjugate, as required. Let a(RG) be the Green ring (see Sec.1 of Chapter 10). Denote by a,(RG) the Z-linear span of all [ V ] where , V is a p-permutation RG-module. Then, by Corollary 1.2(ii), a,(RG) is a subring of a(RG).
Corollary 1.6.
The map
is a ring isomorphism. Proof. It is clear that a,(RG) is a free Z-module freely generated by all [ V ] ,where V runs through the nonisomorphic indecomposable p-permutation RG-modules. A similar statement holds for the subring a,(FG) of a(FG). Hence the desired conclusion is a consequence of Theorem 1.5.
2
Dress’s theorems
Throughout this section, we fix a field F of charactristic p > 0 and a finite group G. If H is a subgroup of G, then any FH-module is assumed to be finitely generated. Our aim is t o provide some further characerizations of p-permutation modules which are contained in a work of Dress (1975). This will be achieved with the aid of the notion of virtually monomial modules defined below. Let V be an FG-module. Then V is said to be monomial if there exists a basis q , . . . ,TJof , V such that G permutes the subspaces F q , . . . ,Fv, of V . Expressed otherwise, V is monomial if and only if V is a direct sum of modules of the form W G ,where W is one-dimensional FH-module for some subgroup H of G. Following Dress (1975), we say that V is virtually monomial if there exist monomial FG-modules V1 and V2 such that
v 69 v,?z v, It is plain that the direct sums of monomial (virtually monomial) modules are again monomial (virtually monomial) modules. Our main result of this section will be proved with the aid of the following theorem which is of independent interest.
p-Permutation Modules
778
Theorem 2.1. (Dress (1975)). Let F be an algebraically closed field of characteristic p > 0 and let V be an indecomposable FG-module with vertex Q and a trivial source. Then there exist subgroups H I , . . . ,H, of G and one-dimensional FH;-module Q, 1 5 i 5 n, such that : @ v,G (for some IC < n). (i) v @ V? @ .. @ v,G 2 @ (ii) Each H i , 1 5 i 5 n, has a normal Sylow p-subgroup P; such that H;/P; is elementary (i.e. a direct product of a cyclic group and a q-group for some prime q ) and such that
vgl
g;P;gzT1C Q for some g; E G
(1 5 i
5 n)
Proof. We apply induction on IQI. First suppose that Q = 1 and let {E;liE I } be the set of elementary subgroups of G. By Corollary 3.5.13, for each i E I , there exist FE;-modules M ; and N ; such that 1~ @ (@iEzM;G) and
Nf
(1)
have the same composition factors (counting multiplicities). Taking into account that for any FG-module W , the FG-module V @ 3W~ is projective, we may tensor both modules in (1) with V t o obtain the following isomorphism of FG-modules :
v @ (@iEI(VEi @ F MilG) 2 @i€Z(VE,@ F NilG Thus, because V E @F ~ M ; and VE,@F N; are projective FE;-modules, we may harmlessly assume that G is an elementary group. If p { ]GI, then there is nothing to prove. We may therefore assume that pl IGl and write G = P x H , where P is a pgroup and H is an elementary p’-group. Let Vl,.. . ,& be all nonisomorphic simple FH-modules. Then it is clear that..,:fI .,KG are all nonisomorphic projective indecomposable FG-modules. Since V is projective, we have V 2 yGfor some i E (1,. .. ,t} and we are again reduced to the case where p { [GI. This establishes the case where Q = 1. Now suppose that IQI > 1 and let H = NG(Q). By the Green correspondence, there exists an indecomposable FH-module W with the same vertex Q and trivial source such that
W G E Uo @ UI @ . . . @ Urn
(U; is indecomposable, 0 5 i 5 m )
with Uo 2 V and IQI < IQil, Q; is a vertex of U; for i E (1,. . . ,m}. Bearing in mind that W is a direct summand of a permutation FH-module, W Gis a
2 Dress’s theorems
779
direct summand of a permutation FG-module. Thus U1,.. . , U,,, are direct summands of permutation FG-modules. Hence, by Theorem 1.1 and the induction hypothesis, the theorem holds for each FG-module U;, 1 5 i 5 m. Consequently, we need only prove the result for W , i.e. we may harmlessly assume that Q d G. Because Q Q G, we see that Q acts trivially on ( 1 ~and ) ~ hence on its direct summand V . Therefore V is inflated from an F(G/Q)-module V , where V is also projective. Consequently, we may apply the case Q = 1 t o infer that V satisfies the conclusion of the theorem (with G replaced by G/Q). This means that there exist subgroups
Hl/Q,...,Hn/Q of G/Q and one-dimensional F(Hi/Q)-modules q, 1 5 i 5 n, such that @ - .. CEI V: (L < n). (a) V @ V p -..@ V f 2 (b) Each H;/Q, 1 5 i 5 n , is elementary of order not divisible by p. Hence, if V , is the FG-module inflated from q,then by (a) and (b), H I , . . . , H , and V,, . . . ,V, satisfy the conclusion of the theorem. This concludes the proof of the theorem. W
Vzl
Specializing to projective FG-modules, we immediately deduce Corollary 2.2. (Dress (1975)). Let F be an algebraically closedfield of characteristic p > 0 and let V be a projective FG-module. Then there exist subgroups HI,. .. ,H , of G and one-dimensional FH;-module V,, 1 5 i 5 n , such that
v @ V? @ - .. CB v,G
2
vLl
..
@
v,G (L < n )
and such that each H i is an elementary p’-group. Proof. We may clearly assume that V is indecomposable, in which case Q = 1 is the vertex of V . Hence the desired conclusion follows from Theorem 2.1. W Another consequence of Theorem 2.1 is given by Corollary 2.3. (Dress (1975)). Let F be an algebraically closed field of characteristic p > 0 . Then there exist subgroups H I , . . . ,If, of G and one-dimensional FH;-module V;, 1 5 i 5 n, such that
p-Permutation Modules
780
--
k+l @ @ V: (for some k < n). (i) 1~@ KG @ * @ VkG - VG (ii) Each H;, 1 5 i 5 n, has a normal Sylow p-subgroup P; such that H;/P; is elementary.
Proof. Let Q be a Sylow p-subgroup of G. Then, by Theorem 6.3.7, Q is a vertex of 1 ~It. is clear that 1~ is a source of 1 ~Hence . the desired assertion follows by virtue of Theorem 2.1. H
The result above allows us t o deduce the following additional property
Theorem 2.4. (Dress (1975)). Let F be a n algebraically closed field of characteristic p > 0. Then, for any FG-module V , there exist subgroups H I , . . . ,H , of G and FH;-modules W;, 1 5 i 5 n, such that : (i) V @ W y @ @ W f 2 WkG tl @ @ W: (for some k < n). (ii) Each H i , 1 5 i 5 n , has a normal Sylow p-subgroup with elementary
---
---
factor group.
Proof. Tensoring the isomorphism in Corollary 2.3(i) with V , we ob-
tain
v @ (@%,(v@F’ yG))
@y=k+l(V@ p KG)
Bearing in mind that, by Theorem 18.5.1 in Vol.1,
the result follows by setting W; = VH~@ F
K, 1 5 i 5 n. W
We have now come the the demonstration for which this section has been developed.
Theorem 2.5. (Dress (1975)). Let F be an arbitrary field of characteristic p > 0 and let V be a nonzero FG-module. Then the following conditions are equivalent : (i) V is a p-permutation module. (ii) V is a direct summand of a monomial FG-module. Moreover, under the assumption that F is algebraically closed, each of the above conditions is equivalent to : (iii) V is virtually monomial.
3 The Brauer morphism
78 1
Proof. Because each permutation FG-module is monomial, the implication (i) + (ii) is a consequence of Theorem 1.1. The implication (iii) + (ii) is a consequence of the definition of virtually monomial modules. Observe also that if F is algebraically closed, then (i) implies (iii) by virtue of Theorems 1.1 and 2.1. By the foregoing we need only show that (ii) implies (i) for an arbitrary F . To do this, we may, by Theorem 1.1, assume that V 2 W G where W is a one-dimensional FH-module for some subgroup H of G. But W is a p-permutation module because a Sylow p-subgroup of H acts trivially on W . Thus, by Corollary 1.2(iv), V is a p-permutation module. This concludes the proof of the theorem.
3
The Brauer morphism
An important tool for the investigation of p-permutation modules is the so called Brauer morphism. Our principal goal here is to record some basic properties of the Brauer morphism with an eye to future applications in the next section. Most of the results recorded are contained in Broud (1985) and Broub and Puig (1980). As usual, we begin by introducing the notation and assumptions. Throughout, p denotes a prime number and R a commutative local ring with
c h a r ( R / J ( R ) )= p We put F = R / J ( R ) and fix a subgroup K of a finite group G. Let V be an RG-module. Given a subgroup H of li, we put
v,l(= Trjy(lnv(VH)) and
V (K ) = l n v ( V ~ ) /
VE
+ J(R)lnv(V~)
where H runs over the set of all proper subgroups of K with the convention that Thus, for Ir' = 1, V ( K )= V / J ( R ) V . It should be pointed out that in most papers on the subject 1 n v ( V ~ is ) denoted by V H ;we shall use this notation only in the case where V = A
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is a G-graded algebra so as not to conflict with our notation for induced modules. Following Brou6 (1985), we refer to the natural surjection
BTL : Inv(Vh.) --+ V ( K ) as the Brauer morphism . We now proceed to record some basic properties of the Brauer morphism.
Lemma 3.1. Let H K be subgroups of G. Then (i) Inv(l/h) is un R(NG(li)/K)-moduZe.
(ii)
( C H C K Vi)+J(R)Inv(Vh)is an R(NG(K)/li)-submod2ze o f ~ n v ( v X ) .
(iii) V ( K ) is an R(NG(K)/K)-module annihilated by J ( R ) . Proof. It is clear that (iii) is a consequence of (ii) and the definition of V ( K ) .If g E NG(K) and v E l n v ( V ~ )then , for any z E K, z(gv) = g(g-lzg)v = gv
proving that I n v ( v ~ is ) an R(NG(li))-module. Because li’ acts trivially on Inv(Vh),it follows that Inv(Vh-) is an R(NG(K)/li)-module in a natural way. This proves (i). We are left to verify (ii). If K = 1, there is nothing to prove. Suppose that H is a proper subgroup of K and let T be a left transversal for H in Ii. Given v E 1 n v ( V ~ )and g E N ~ ( l i >we, then have
= TrgHg-I(gv) K E
vSg-,
as required.
Corollary 3.2.
For any subgroup K of G, the Brauer morphism BrK : ~ n v ( ~ + h )V ( K )
is a homomorphism of R( NG(K)/K)-VZOdUkS Proof. This is a direct consequence of Lemma 3.1. W
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783
Next we investigate the behaviour of the R(NG(I<)/K)-module V ( K ) .
Lemma 3.3. Let Ii' be a subgroup of G. Then (i) If V = U @ W for some RG-modules U and W , then V(1i) E U ( K )@ W ( K ) as R(NG(K)/K) - modules
(ii) v ( I { ) = vN,(,)(K). (iii) If K acts trivially on V and K is a p-group, then V ( K )= V/J(R)V Proof. (i) For any subgroup H of h', we have
I n v ( v ~=) l n v ( U ~@) Inv(WH) Taking into account that
Tr$(Inv(UH) @ 1 n v ( W ~ )=) TrE(Inv(UH))@ Tr;(lnv(WH)) the desired assertion follows. (ii) This is a direct consequence of the definition of V ( K ) . (iii) Assume that K acts trivially on V and Ii is a p-group. If 11' = 1, then by convention V ( K )= V / J ( R ) V . Hence we may assume that Ii # 1. Then, for any proper subgroup H of K , we have
V[
c J(R)Inv(V,)
since char(R/J(R))= p. Taking into account that I n v ( r / i )= V , we therefore deduce that V ( K )= V / J ( R ) V as desired. Next we examine how the Brauer morphism behaves with respect to relative trace maps
Lemma 3.4. For any subgroup K of G and any RG-module V , we
Proof. (i) The first equality is a consequence of the fact that li acts trivially on V ( K ) .Thus we need only show that for all v E Inv(VK),
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Let T be a full set of double coset representatives for ( N c ( K ) , K ) in G containing 1. Then, by Lemma 18.8.6(i) in Vol.1,
Kext we fix t E T - {l},put L = tKt-' n Nc(li) and denote by S a full set of double coset representatives for ( K ,L ) in N c ( K ) . Then each SLS-' n K is a proper subgroup of K and so, by Lemma 18.8.6(ii) in Vol.1,
TrP(")(tv) E
c
C T rfis-l nK( I n v (VsLs-inK)) SES
Iier(BrK)
This establishes (l),by applying (2). (ii) This is a direct consequence of (i). Assume that U , V and W are RG-modules. An R-bilinear map
f:UxV+W is said t o be G-stable if
f ( g u , g v ) = g f ( u ,w ) for all g E G, u E U , v E V The following lemma provides some properties of G-stable bilinear maps. Lemma 3.5. Let U V and W be RG-modules and let f : U x V be a G-stable R-bilinear map, For each u E U , v E V , define
by the rule :
Then the maps
are RG-homomorphisms.
---f
W
3 The Brauer morphism
u E
785
Proof. It is plain that both maps are R-linear. Kow fix g E G and U . Then, for all u E V ,
proving that f g u = g f u . Thus u +, f u is an RG-homomorphism. A similar argument shows that fg,, = g f v for all g E G , u E V . Thus u I+ f v is also an RG-homomorphism. The main property of G-stable R-bilinear maps is given by the following result.
Theorem 3.6. Let U , V and W be RG-modules and let f:UxV-+W
-
be a G-stable R-bilinear map. Then, f o r any subgroup I< of G, the map fK : U ( K )x
V(K)
W(Ii)
defined by fh.(BrK(u),Brj&)) = BTF( f ( U , Q ) ) 1 n u ( U ~ )u, E 1 n v ( V ~ )is) a well-defined ( N ~ ( l < ) / I i ) - s t a bR-bilinear le map.
(u E
Proof. Given u E 1 n v ( U ~and ) TJ E Inu(VK), it follows that for all g E K, s f ( u , 4 = f b , 9 4 = f ( u ,4 which shows that f ( u ,u) E Inv(W,v). To prove that fK is well-defined, it suffices to verify that if u E K e r ( B r K ) or u E K e r ( B r K ) ,then B r F ( f (21, u)) = 0 Assume that u E K e r ( B r g ) . Then
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for some X E J(R), x E I n v ( U k ) , U H E U z , where H runs over the set of proper subgroups of ' A (if K = 1, then put all U H = 0). Thus H
and therefore it suffices to show that B r E ( f ( U H , v)) = 0 for all H C K. Now U H = &T t y for some y E l n v ( U ~ )where , T is a left transversal for H in I<. Because v E Inv(VK), y = l n v ( U ~and ) f is G-stable, we have f(y,v) E I ~ ~ ( W H Hence ).
f ( U H v) 7
= Cf(ty,v) 1€T
= Ctf(y,v) t€T
= T r 3 f (y, 4) E Ker(BrF)
A similar argument shows that if v E l < e r ( B r i ) , then B r z ( f ( u , v ) )= 0 and thus f is well-defined. It is plain that f is an R-bilinear map. Moreover, for any g E N G ( K ) , we have
fh.(sBrm,sBra4) = = = =
fK(BTK(94,
NkP))
~rF(f(9~7gz.)) sBr.Kw(f(., 4) S f K ( B 6 2 4 ,B r 2 4 )
by applying G-stability o f f together with Corollary 3.2. This completes the proof of the theorem. 1 Now we concentrate on permutation modules. Theorem 3.7. Let V be an RG-module, let P be a p-subgroup of G such that V p is a permutation RP-module with an R-basis X permuted by P , and let V = V / J ( R ) V . Denote by 2 1 , . . . , x , all distinct elements of X fixed by P . Then the following properties hold : (i) The set {Br;(x;)Il 5 i 5 n } is an F-basis of V ( P ) . (ii) The F(NG(P)/P)-modulesV ( P ) and V ( P ) are canonically isomorphic.
3 The Brauer morphism
787
(iii) If V * is the contragredient of V and the basis X * = { X * ~ X E X } of V * is dual to X , then there is an F(NG(P)/P)-isomorphism V*(P)
---f
V(P)*
which sends the basis {Br$*(z*)ll5 i 5 n } of V * ( P )onto the dual basis of { ~ r ; ( z L . ; )5 [ l i 5 n}. Proof. (i) Let q , z 2 , . . . ,z n , %,+I,, . . ,z t be all representatives for the P-orbits of X , and let Q; be the stabilizer of 2; in P , 1 5 i 5 t. Due t o our choice of xi, we have = Q2 =
Q1
* * a
= Qn = P
and each Qn+l,. . . ,Qt is a proper subgroup of P . Owing to Lemma ll.l.l(iii),
{Tr&)ll
I: i 5 t }
is a n R-basis for Inv(Vp). Thus we need only verify that for any proper subgroup Q of P , VQ' is contained in M , where hl is the sum of J ( R ) l n v ( V p ) and the R-linear span of T r g b ( z k ) ,n 1 5 k 5 t. To this end, fix z E X and denote by H and L the stabilizers of x in Q and P,respectively. Because the elements v E V of the form v = T rQH ( z ) form an R-basis for l n v ( V Q ) ,we need only verify that for any such v, we have T r g ( v ) E M . Taking into account that
+
TTQP(TTS(.))
= Trf;(z)
=
TTI(TTfj(Z))
and T r f i ( z )E J ( l l ) I n v ( V ~whenever ) L # H , the required assertion follows. (ii) We first observe that V p is a permutation FP-module with F-basis
x = (312 E X } where 3 = z
+ J ( R ) V . Therefore, by (i), {B~:(Z;)~I 5 i 5 n }
is an F-basis of V ( P ) . Thus
V(Y) BrpV(z;)
+ H
V(p)
BrF(3;)
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is an F-isomorphism. Because this isomorphism clearly preserves the action of N c ( P ) / P ,the required assertion is established. (iii) Let a, b E X and let ga = b for g E P . Then ga*(b) = a*(g-lb) = a*(a) = 1 and, for
P
# b, P
EX,
ga*(x) = a*(g-lx) = o
Consequently, ga* = b* and similarly if ga* = b*, then ga = b. In particular, it follows that x;, . . . ,x; are all distinct fixed points of X * in P. Thus, by (i), {Br;*(xf)ll 5 i 5 n} is an F-basis of V * ( P ) . We deduece therefore that the map
{
V * ( P ) 4 V(P)* B T ; * ( ~ ? )H (BrpV(x;))*
is at least an F-isomorphism. Now let f : V x V* R be defined by f(v,cp) = ~ ( v for ) all v E V , y E V'. Then, regarding R as the trivial RG-module, f becomes a G-stable R-bilinear map. Because R ( P ) = R/J(R) = F , it follows from Theorem 3.6 that the map --f
x V * ( P )-+ F
fp : V ( P )
defined by
f P ( B r 2 4 ,Brg*(cp))= cp(v>t J(R) (v E Inw(Vp),cp E Inw(V,')) is an (Nc(P)/P)-stable F-bilinear map. Thus, by Lemma 3.5, the map
{
V*(P) B.pV*(cp>
-+
V(P)* fv
where fv(Br;(w)) = ~ ( vt) J ( R ) ,is an F(Nc(P)/P)-homomorphism. Setting y = xr, we see that this homomorphism coincides with $ and the result is established. We now take a close look at the module V ( I i ) ,where I i is any subgroup of G and V is an H-projective RG-module, for some subgroup H of G.
Lemma RG-module. conjugate to W G ( K )= 0
3.8. Let H be a subgroup of G and let V be an H-projective Then, for a n y subgroup I i of G, V ( K ) = 0 unless Ii is Ga subgroup of H . In particular, if W is any RH-module, then unless A' is G-conjugate to a subgroup of H .
3 The Brauer morphism
789
Proof. Because V is H-projective, it follows from Theorem 18.9.8 and Lemma 18.9.6(ii) (both in Vol.1) that
1v
E
c
TrE(EndRH(V)) T r E n g H g - l (EndR(KngHg-l)(V)) g€G
For each g E G , we may therefore choose 1~ =
& E E n d R ( K n g ~ j g - ~ )such ( V ) that
K C TrKngHg-1
($9)
SEG
and therefore
However, for any v E Inv(V-), we have
thus completing the proof.
) , ~ R His the Our next aim is to compute explicitly ( l ~ ~ ) ' ( pwhere trivial RH-module and P is ap-subgroup of G . This will be achieved with the aid of Lemma 3.8. Note that, by Lemma 3.8, we may harmlessly assume that P is G-conjugate to a subgroup of H . Theorem 3.9. Let P be a p-subgroup of G , let H be a subgroup of G such that P is G-conjugate to a subgroup of H and let T be a set of all representatives t for the double cosets NG(P)tH for which P C tHt-'. Then
as F(NG(P)/P)-modules.
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Proof. Let X 2 T be a full set of representatives for ( N G ( P ) , H ) double cosets in G. Then, by Mackey decomposition,
where V, is defined by
On the other hand, by Lemma 3.8,
Moreover, if P E z H x - ~ ,then P acts trivially on V, and so, by Lemma 3.3(iii), we have
The desired conclusion is now a consequence of (3), (4), (5) and Lemma 3.3(i), (iii). H We now apply the Brauer morphism to a very important special case where the RG-module V is some G-algebra A. First, we introduce some notational conventions. Given a G-algebra A over R and subgroups H C 'h of G, we put
where A H is the subalgebra of H-invariant elements of A and
TT; : AH --i A" is the relative trace map. Next we observe that
3 The Brauer morphism
791
It follows from (6) and (7) that A; "(AH)
is an ideal of AK. Observe also that
= ASH9-I
for all g E G
(8)
and that
Because the G-algebra A is an RG-module, we may define A ( K ) as we did for any RG-module V , i.e. by putting
A ( K ) = A"/
L
C A E + J(R)A"
where H runs over the set of all proper subgroups of that = 0 for K = 1
) K with the convention
EAi H
In particular, the natural surjection
B r i - : AK
+
A(li)
gives us the Brauer morphism with respect t o the RG-module V = A . Lemma 3.10. Let li be a subgroup of G and let A be a G-algebra over
R. Then
(i) l i e r ( B r ; ) is an ideal of A". (ii) A" is an (NG(li)/li)-algebraover R, while A ( K ) is an ( N G ( K ) / K ) algebm over R annihilated by J ( R ) . (iii) The Brauer morphism
B r i : A"
-, A ( K )
is a homomorphism of (NG(lr')/K)-algebras,i.e. a homomorphism of Ralgebras preserving the actions of N G ( K ) / K on AK and A(1i). Proof. (i) This follows directly from the fact that both A; and J(R)A" are ideals of A". (ii) The assertion concerning A" follows from (8) applied to H = K . The remaining assertion is a consequence of (9) and the definition of A ( 1 i ) . (iii) Apply (i) and Corollary 3.2. H
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792
We close by providing the following observation which justifies the term “Brauer morphism”.
Lemma 3.11. Let A = R X be the group algebra of a group X over R and let G act on X as a group of automorphisms of X . Then A becomes a G-algebm in the natural way and the following two properties hold : (i) If P is a p-subgroup of G and H is the subgroup of all elements of X fixed by P, then the map
{
CheHhBr$(h)
--f
FH
I+
ChcHAhh
(Ah E
F)
is an isomorphism of F-algebras. (ii) If R = F , X = G and G acts on itself by conjugation, then H = CG(P) and, upon the identification of FG(P) with FCG(P), the restriction of BT;’ to Z ( F G ) is the classical Brauer homomorphism (see Sec.3 of Chapter 8 in Vol.1).
Proof. (i) This follows from Lemma 3.lO(iii) and Theorem 3.7(i). (ii) Let C1,. .. ,C,, be all P-conjugacy classes of G. If ICil > 1, then BTFG(C?) = 0
while, for all h E CG(P),
BrFG(h)= h
This clearly implies the desired assertion.
4
Scott modules
In what follows, we fix a prime number p and a finite group G. We also assume that R is either a complete discrete valuation ring with c h a r ( R / J ( R ) )= p or R is a field of characteristic p , and put
F = R/J(R) All RG-modules are assumed t o be finitely generated. To emphasize the coefficient ring R, we write 1 ~ ’for the trivial RG-module.
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793
Our principal goal is to examine an important type of indecomposable p-permutation RG-modules, the so called Scott modules . Such modules were first discovered by L.L. Scott (unpublished) and the first published account can be found in Burry (1982). According to Burry (1982, p.1856), Scott modules were later rediscovered by Alperin. Our method below is based entirely on a work of Brou6 (1985) who utilized a systematic use of the Brauer morphism. All the relevant background concerning Brauer morphisms can be found in the preceding section. We begin by introducing the following definition due to Puig (see Brou6 (1985)). Let V be a p-permutation RG-module and let P be a p-subgroup of G. Then the Scott coefficient of V associated with P , written sp(V) is defined by sp(V) = dimF(BrpV(V,G)) where
Br,v : Inv(Vp)
---f
V(P)
is the Brauer morphism introduced in Sec.3 and
Observe that, by Lemma 3.4(ii) applied t o K = P , we have
Our first task to investigate the behaviour of s p ( V ) . Lemma 4.1. Let P be a p-subgroup of G. (i) If V and V' are two p-permutation RG-modules, then
sp(V @ V')= S P ( V )
+
SP(V')
(ii) If V is a p-permutation RG-module and V* is the contragredient of
V , then
+(V) = SP(V*)
(iii) For any subgroup H of G, s p [ ( l ~ ~= )1~if ]P is G-conjugate to a Sylow p-subgroup of H and s p [ ( l ~ ~ ) = ' ] 0 otherwise. Proof. (i) Apply Lemma 3.3(i) and (1). (ii) Invoking Theorem 3.7(iii), we see that sp(V*) is equal t o the rank of
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the endomorphism T r NG(p)’p , of V(P)*. The latter is equal t o the rank of its transpose, i.e. of the endomorphism T r N , ~ ( p ) ’ pof v ( P ) . Consequently,
w(V) = V ( V * ) as desired. (iii) Owing t o Lemma 3.8, ( ~ R H ) ~=(0Punless ) P is G-conjugate to a subgroup of H. By Sylow theorems in H ,if g-l P g is a Sylow p-subgroup of H for some g E G,then g r l P g l is a Sylow psubgroup of H only for those g1 E G for which g1 E NG(P)gH. Invoking Theorem 3.9, it therefore suffices t o show that has dimension 1 or 0 according t o whether H is , or is not, a p’-group. By Lemma 11.1.3(ii), we may identify ( ~ F H with ) ~ the submodule FG. H+ of FG. Then we have
((1~~)~)7
(( ~
F H ) ~ )= ?
TrF(FG
*
P)
= G+.FG.H+ = FG.G+.H+ = IHI(FG+),
as desired. We can now obtain our desired goal.
Theorem 4.2. Let P be a p-subgroup of G. Then (i) There exists a unique (up to RG-isomorphism) indecomposable ppermutation RG-module Sp(G,R) such that sp(Sp(G,R ) ) # 0. Moreover, sp(Sp(G,R)) = 1, and
SP(G, R) 2 [Sp(G, R)]* (ii) If H is a subgroup of G, then Sp(G, R ) is isomorphic to a direct summand of ( 1 ~if and ) ~only if P is G-conjugate to a Sylow p-subgroup of H . In particular, by taking H = P, S p ( G ,R) has vertex P.
Proof. (i) Owing t o Theorem l.l(vi), any indecomposable p-permutation RG-module V is a direct summand of ( I R Q ) for ~ some p-subgroup Q of G. If s p ( V ) # 0, then by Lemma 4.1(i), (iii), we see that P is G-conjugate to Q . Moreover, since s p ( ( l ~ p ) = ~ )1, it follows that V is the unique indecomposable direct summand of ( 1 ~ psuch ) ~ that s p ( V ) # 0. Hence s p ( V ) = 1
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795
and, by the unicity of V , it follows from Lemma 4.l(ii) that V 2 V'. (ii) It is a consequence of (i) and Lemma 4.l(i) that Sp(G, R ) is isomorphic t o a direct summand of ( ~ R H ) ' if and only if
#0
sP((1RH)')
Thus, by Lemma 4.l(iii), P is G-conjugate t o a Sylow p-subgroup of H if and only if S p ( G , R ) is isomorphic t o a direct summand of ( ~ R H ) This ~ . concludes the proof of the theorem. W We shall refer to the RG-module Sp(G,R) as the Scott module of G associated t o P . The aext result provides some further properties of Scott modules.
Let P be a p-subgroup of G and let H be n subgroup of G such that P is G-conjugate to a Sylow p-subgroup of H. Then (i) H O ~ R G ( ~SP(G, R G ,R ) ) 2 H o ~ R G ( S P ( G R ), ,~ R G 2 ) R. (ii) Sp(G, R ) is the unique indecomposable direct summand V of ( ~ R H ) ' such that H O ~ R G ( ~VR) # G 0, Theorem 4.3.
(iii) Sp(G,R ) is the unique indecomposable direct summand V o f ( 1 ~ ~ ) ' such that HomRG(V, 1 R G ) # 0 Proof.
Owing to Proposition 18.1.6 in Vol.1, we have HomRG(lRG, (lRP)G)
HOmRP(1RP, 1 R P )
S
R HomRG'(((lRP)G,lRG)
Since, by Theorem 4.2(i), S p ( G ,R ) 2 Sp(G, R)* it suffices to show that HomRG(lRG, SP(G, R ) ) # 0
(2)
To this end, we observe that, by Lemma 3.4(ii), (Sp(G,R))$ is mapped onto ( S d G ,R ) ) ( P N ) ,G ( p ) ' p ; this last module is not zero by the definition of Sp(G, R ) because its dimension is precisely sp(Sp(G, R ) ) . Thus
W S P ( G , R ) )# 0 which establishes (2). This concludes the proof of the theorem. H
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Next we relate S p ( G , R ) with Sp(G,F) and examine t o what extent S p ( G ,R ) determines P.
Theorem 4.4. Let P and Q be p-szlbgroups of G. Then (i) SP(G,R ) / J ( R ) S P ( GR, ) &(G, F ) . (ii) SQ(G,R ) S Sp(G,R ) if and only if Q is G-conjugate to P . Proof. (i) Let V be any ppermutation RG-module and let
v = V / J (R)V Owing to Theorem 3,7(ii), we have sp(V) = s p ( v ) . Thus by the characterization of Sp(G, R ) given by Theorem 4.2(i), we deduce that
Sp(G,R )
S d G ,F)
as required. (ii) This is a direct consequence of Theorem 4.2(ii). H
The next result examines two extreme cases for R = F .
Corollary 4.5. Let P be a Sylow p-subgroup of G . Then (i) S p ( G , F ) ~ F G . (ii) S I ( G , F ) is a projective cover of ~ F G . Proof. (i) Apply Lemma 11.1.3(v) and Theorem 4.3(ii). (ii) Apply Theorem 4.3(iii). H The rest of this section is based on a work of Burry (1982). In what follows, F denotes an arbitrary field of characteristic p > 0. Let us examine the Green correspondents of Scott modules.
Theorem 4.6. Let P be a p-subgroup of G , let V = Sp(G,F) and let H = N G ( P ) . Then the Green correspondent f ( V ) of V with respect to ( G ,P , H ) can be considered as an F(Ei/P)-module and as such f ( V ) is a projective cover of l F ( H / p ) . Proof. By Theorem 4.2(ii), P is a vertex of V . Hence f ( V ) is indeed defined. Since the Green correspondence preserves vertices and sources, we
4 Scott modules
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see that P is a vertex of f ( V ) and 1 ~ ispa source of f ( V ) . Thus f ( V ) is a component of ( 1 ~ p ) But, ~ ~ since . P a H , we have ( 1 p p ) H 2! F ( H / P )
by virtue of Theorem 18.5.1 in Vol.1. Thus f ( V ) can be regarded as an F(H/P)-module and as such f ( V ) is a projective indecomposable module. On the other hand, since
0
#
HOmFG(lFG,V)
C HomFG(lFG, f(V)')
(by Theorem 4.3(i)) (by Theorem 7.2.1)
HomFH(lFH,f(V)) (by Proposition 18.1.6(ii) in Vol.1) it follows that the FH-module f (V) has a trivial submodule. Hence the same is true for the F(H/N)-module f(V). This shows that Soc( f(V)) 2 l p ( ~ / q and hence f(V) is a projective cover of l ~ ( ~ / pH) . Assume that H is a subgroup of G and P is a p-subgroup of H . Then P determines two Scott modules, namely
It is therefore appropriate to investigate whether the above modules are related t o each other. The next result provides some pertinent information.
Theorem 4.7. Let H be a subgroup of G and let P be a p-subgroup of H. Then the following properties hold : (i) Sp(G,F) is a component of S P ( H , F ) ~ . (ii) Sp(H, F ) is a component of [Sp(G,F ) ] H .
Proof. (i) Put Q = N H ( P ) / Y , L = N c ( P ) / P , U = S p ( G , F ) and V = S p ( H , F ) . Let X be the Green correspondent of V with respect to (H,P,N,(P)) and let Y be the Green correspondent of U with respect to
(G, P, NG(P)).
Owing t o Theorem 4.6, X can be regarded as an FQ-module and as such X is a projective cover of ~ F QSimilarly, . Y can be regarded as an FL-module
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798
and as such Y is a projective cover of ~ F L . Now there is a surjective F Q homomorphism X + ~ F and Q so there is a surjective FL-homomorphism of induced modules : X L (1FQY On the other hand, since
H o ~ F L ( ( ~ F Q ) ~2, ~ HFOL~)F Q ( ~~FFQQ,#) 0 there is a surjective FL-homomorphism (lFQ)L
+
1FL
Thus there is a surjective FL-homomorphism
xL + 1 F L Since X L is a projective FL-module and Y is a projective cover of ~ F L it, follows from Proposition 7.1.2 in Vol.1 that Y is a component of X L . Hence the FNG(P)-module Y is a component of the F(Nc(P))-module XNG('). This proves (i), by applying Theorem 7.2.5(ii). (ii) There is a surjective FL-homomorphism Y -+ ~ F and L so there is a surjective FQ-homomorphism YQ + ~ F Q .Since YQ is a projective F Q module, i t follows from the argument in (i) that the F(Nff(P))-module X is a component of Y N H ( p ) .This proves (ii), by applying Theorem 7.2.5(i). H Note that Theorem 4.7(i) tells us that if we induce a Scott module with vertex P , we get a Scott module with vertex P as a component. The next theorem is a converse of this. More precisely, we have the following result. Theorem 4.8. Let H be a subgroup of G and let P be a p-subgroup of H . If U is an indecomposable p-permutation RH-module and Sp(G, R ) is a component of U G , then U 2 Sgpg-l( H ,R ) , for some g E G with gPg-' H.
Proof. By Theorem 1.1, U is a component of ( ~ R L for ) some ~ subgroup L of H . Setting S = Sp(G,R ) , it follows that S is a component of ( ~ R L ) ~ . Hence, by Theorem 4.2(ii), P is G-conjugate to a Sylow p-subgroup, say Q, of L. Moreover, since 0
#
c 2
HOmRG(lRG,S) HomRG( 1RG, uG)
H O ~ R H~ R( H u , )
5 Vertices of p-permutation modules
799
it follows from Theorem 4.3(ii) (with W = L , G = H and P = Q ) that U S S Q ( H , R ) . This completes the proof of the theorem. W
5
Vertices of p-permutation modules
We preserve all notation and conventions introduced in the previous section. Our aim is t o investigate vertices of p-permutation modules. This will be achieved by applying a number of properties of the Brauer morphism t o ppermutation modules. Our point of departure is the following two auxiliary assertions, both extracted from a work of Broud (1985).
Lemma 5.1. Let P be a p-subgroup of G and let V be a p-permutation RG-module. Then V ( P ) is a p-permutation F(NG(P)/f')-module. Proof. Denote by Q a Sylow p-subgroup of N G ( P ) .Then, by Theorem 1.1, VQ is a permutation RQ-module. Choose an R-basis X of V which is permuted by Q. Then the F-basis for V ( P ) given by Theorem 3.7(i) is permuted by Q / P , as asserted. H
Lemma 5.2. Let P be a p-subgroup of G , let V be a p-permutation RG-module, and let A = EndR(V). Then the map
where
is an isomorphism of F(Nc(P)/P)-algebras. Proof. The natural bilinear map f : A x V G-stable. Thus, by Theorem 3.6, the map fp : A ( P )x
V ( P )--+ V ( P )
--+
V , (a,v) H a(.)
is
p-Permutation Modules
800
is a well-defined (NG(P)/P)-Stable F-bilinear map. Therefore, by Lemma 3.5, the given map is a homomorphism of F(NG(P)/P)-algebraS. Let X denote an R-basis of V which is permuted by P. Then the set
where ax,y E A is defined by
al,y(z) = Sy,*x
for all
2,
y,z E X
is an R-basis of A which is permuted by 1'. Let C x ( P )be the set of fixed points of X under P . Then we clmrly have cX(A)(P)
= {ax,ylx, Y E c X ( P ) }
Given z E Cx(P)and a E C X ( A ) ( P )we , now put Brj!(a). Then we must have
a:
= B r F ( x ) and u =
It therefore follows from Theorem 3.7(i) that the given homomorphism is an isomorphism. H We are now ready to investigate vertices of p-permutation modules.
Theorem 5.3. (Broue' (1985)). Assume that V is a n indecomposable p-permutation RG-module. Then (i) The vertices of V are the maximal p-subgroups P of G such that V ( P ># 0. (ii) A p-subgroup P of G is a vertex of V if and only if V ( P ) # 0 and V ( P ) is a projective F(NG(P)/P)-module. Proof. (i) Assume that a p-subgroup P of G is a vertex of V . Then, by Lemma 3.8, it suffices to verify that V ( P ) # 0. We know that V is a component of ( 1 ~ p ) Hence, ~ . by Lemma 11.1.9, Vp is isomorphic to a direct sum of modules of type : [lR(PrlgPg-')I
P
where g runs over a certain subset S of G. But V is a component of and because P is a vertex of V it follows that S n NG(P) # 0. Thus 1 ~ isp a component of V p and so V ( P )# 0.
5 Vertices of ppermiitatiori modules
801
(ii) Suppose that V ( P ) # 0 and that V ( P ) is a projective F ( N G ( P > / P j module. Setting A = E n d R ( V ) , it follows from Lemma 5.2 and Theorem 18.9.8 in Vol.1 that A(p)NC(")Ip = ( A ( P ) ) F ( p ) / p But, by Lemma 3.4(ii), ( A ( P ) ) ,N G ( p ) ' p is the image of A: under the Brauer morphism Br;?. Because l v is the unique nonzero idempotent of AG, we deduce that l v E A:, proving that V is P-projective. Hence, by (i), P is a vertex of V . Conversely, suppose that P is a vertex of V . Then l v E A: and therefore
Hence, by Lemma 5.2, V ( P ) is a projective F(NG(P)/P)-module. H Next we enumerate all indecomposable p-permutation modules with vertex P via the isomorphism classes of projective indecomposable F ( N G ( P ) / P ) modules.
Theorem 5.4. (Broue' (1985) and Klyachko (1979)). Let V be an indecomposable p-permutation RG-module and let P be a p-subgroup of G. Then the map V H V ( P ) induces a bijective correspondence between the isomorphism classes of indecomposable p-permutation RG-modules with vertex P and the isomorphism classes of projective indecomposable F ( NG(P)/P)modules. In particular, S d G ,W ( P )
is the projective cover of lF(NG(P)IP)
Proof. Owing to Theorem 3.9, we have ( ( l R P ) G > ( P ) (IF) N o ( P ) I P
The indecomposable p-permutation RG-modules with vertex P correspond to the indecomposable components of ( 1 ~ p with ) ~ vertex P. Observe also that the projective indecomposable F(NG(P)/p)-modules correspond to indecomposable components of
p-Permutation Modules
802
Now put A = EndR((lRp)G). Then, by Lemma 5.2,
as F(NG(P)/P)-dgebras. The indecomposable components of ( l R p ) G with vertex P correspond to the primitive idempotents of AF whose image in A(P) is nonzero. Because, by Lemma 3.4(ii), B r i maps A$ onto A(P),N G ( P ) / P
and AG = A;, we have
Thus the first assertion follows from the standard facts about lifting idempotents : an indecomposable component V of (lRp)G with vertex P corresponds to a primitive idempotent e of A: such that Brj!(e) # 0, which corresponds to the primitive idempotent B r $ ( e ) of ( A (P ) ) N G ( P ) / P , which in turn corresponds to the direct summand
of ( ~ F ) ~ G ( ' ) / ~The . assertion regarding Sp(G,R)follows from the fact that, by definition, ( S p ( G ,R)(P))NG(P)/P # 0 (alternatively, it follows from the equality S p ( G ,R ) ( P )= Sl(NG(P)/P,F ) .
Corollary 5.5. Suppose that P is a p-subgroup of G . Then the number of nonisomorphic indecomposable p-permutation RG-modules with vertex P is equal to the number of F-conjugacy classes of p-regular elements of
/ p.
NG ( P)
Proof. Apply Theorem 5.4 together with the Witt-Berman's theorem (Theorem 17.5.3 in Vol.1). The following consequence of Theorem 5.4 provides an alternative proof of Theorems 1.4(i) and 1.5(i). Corollary 5.8. The reduction modulo J ( R ) provides a bijection between the isomorphism classes of p-permutation RG-modules and the isomorphism classes of p-permutation FG-modules.
5 Vertices of p-permutation modules
803
Proof. Owing to Theorem 3.7(ii), V ( P )2 v ( P ) where V and P are as in Theorem 3.7. The desired assertion is therefore a consequence of Theorem 5.4. H Another interesting consequence of Theorem 5.4 is given by
Corollary 5.7. (Bmue' (1985)). There is a bijection between the isomorphism classes of indecomposable p-permutation RG-modules and the Gconjugacy classes of pairs ( P , E ) , where P is a p-subgroup of G and E is a projective indecomposable F( NG( P)/P)-module. Proof. This is a direct consequence of Theorem 5.4. H We now probe more deeply into the nature of the correspondence V V ( P )given by Theorem 5.4.
I+
Theorem 5.8. (Bmue' (1985)). Let P be a p-subgroup of G , let M be Q p-permutation RG-module and let E be a projective indecomposable F ( N G (P ) / P)-module. If N is the corresponding indecomposable p-permutation RG-module with vertex P , then N is a component of M if and only i f E is a component of M ( P ) .
Proof. Note that, by definition, the RG-module N is the indecomposable p-permutation RG-module determined by the condition N ( P ) = E . It is plain that if N is a component of M , then E is a component of M ( P ) . Conversely, assume that E is a component of M ( P ) , and put A = E n d ~ ( A 4 )Then, . by Lemma 5.2, A ( P ) S EndF( M ( P ) )
contains a primitive idempotent e such that e M ( P ) E E . Using the standard facts about lifting idempotents together with Lemma 3.4, it follows that A: contains a primitive idempotent f such that B r $ ( f ) = e. Thus f - M is an indecomposable component of M with vertex P , and (f - M ) ( P ) = e . M ( P ) 2 E . This implies that f . M 2 N and so N is a component of M , as desired. H
p-Permutation Modules
804
The next result demonstrates that V ( P ) is a familiar classical object for the case where V is an indecomposable p-permutation FG-module with vertex P.
Proposition 5.9. (Broue‘ (1985)). Let V be a n indecomposable ppermutation FG-module with vertex P . Then the Green correspondent of V with respect to (G,P,Nc(P)) is the FNc(P)-module V(P). Proof. Let W be the Green correspondent of the FG-module V with respect t o (G, P, N c ( P ) ) . Since, by Theorem 1.1, V has a trivial source, W must also have a trivial source. Thus W is isomorphic to a direct summand of (lFP)NG(P)
and so P acts trivially on W , which forces W ( P )= W . But, by Theorem 7.2.1, VNG(P)2 W 63 W’ where W’ is a direct sum of indecomposable F(Nc(P))-modules with vertex strictly contained in P. It therefore follows from Lemma 3.8 that W ’ ( P )= 0. Thus we must have
E
W(P)
(by Lemma 3.3(i))
= w as desired. H Some further interesting properties of p-permutation modules can be found in a work of Klyachko (1979).
6
Puig’s theorem
Throughout, G denotes a finite group and F an arbitrary field of prime characteristic p. All FG-modules are assumed to be finitely generated. Given a p-permutation FG-module V , we may form two NG(P)-algebras
6 Puig’s theorem
805
(all relevant information and notation is contained in Sec.3). The importance of the FNc(P)-module V ( P )stems from the fact that if V is indecomposable with vertex P , then V ( P )is the Green correspondent of V with respect to (G, P, NG(P)) (see Proposition 5.9). We remind the reader that, by Lemma 5.1, V ( P )is also a p-permutation F(Nc(P)/P)-module. Our principal god is to prove that the NG(P)-algebras in (1) are identifiable. This will be achieved with the aid of the following result. Proposition 6.1. (Cabanes (1988)). Let P be a normal p-subgroup of G and let V be a p-permutation FG-module. Let U be the direct sum of those indecomposable direct summands of V whose vertex contains P and let W be the sum of the others. Then (i) V = U @ W , V ( P )2 U and W ( P )= 0 . (ii) I n v ( U p ) = U and I n v ( W p ) = Ker(Br:). (iii) For A = E n d F ( V ) , we have
Proof. (i) and (ii). It is plain that V = U @ W .Moreover, by Theorem 5.3(i), W ( P ) = 0 and, by the definition of U , Inv(Up) = U . Because I n u ( U p ) = U , we have K e r ( B $ ) = 0 (hence U ( P ) = U ) , while because W ( P )= 0 we have K e r ( B r y ) = I n v ( W p ) . Consequently,
V(P)
U(P)$W(P) 2 U(P)=U
and
which proves (i) and (ii). (iii) Let Q be a proper subgroup of Y and let f E EndFQ(V).To prove “E” containment, it suffices to verify that
TrQP(f)(u)E TV Because u E Inv(Vp), we have
for all
uE
u
p-Permutation Modules
806
with f(u) E I ~ V ( V Q But ) . then
TrC(f(4) E v,p g Iier(BrpV) = I n v ( Wp ) as required. To establish the reverse containment, let e : V -+ U be the projection map, and let 1- e = Cy.l e; be the decomposition of 1- e associated with a decomposition of W as a direct sum of indecomposable FG-modules whose vertices do not contain P. Let Pj be a vertex of ej(W),1 5 i 5 n. Then P; does not contain a G-conjugate of P and ej E A:,, 1 5 i 5 n. However, if i~ {1,..., n } , then
gPjg-' IIP
cP
for all g E G
which forces ei E A:,
G
c
C~rp*~-Inp g€G
Iier(Br$)
and therefore 1 - e E Ker(Br$). Thus, i f f E E n d ~ p ( Vwith ) f(U)C W, then ef e = 0 and
f as desired.
+
= (1 - e ) f e f ( 1 - e ) E Ker(Br$)
m
We are now ready to achieve our goal. The following theorem of Puig can be found in a work of Cabanes (1988).
Theorem 6.2. Let V be a p-permutation FG-module and let P be a p-subgroup of G. Then
( E n d F ( V ) ) ( PE ) EndF(V(P))
US
N G ( P )- algebras
Proof. We first observe that Inv(Vp) and K e r ( B r F ) are stable under every element of E n d ~ p ( V ) .Hence the natural action of E n d ~ p ( Von ) V ( P )induces a homomorphism II,: EndFp(V)+ E n d ~ ( v ( P ) )
6 Puig’s theorem
807
of Nc(P)-algebras. It will next be shown that this homomorphism is surjective and has the desired kernel. Write VNG(p)= U @ W , where U is the direct sum of those indecomposable direct summands of V ~ ~ ( whose p 1 vertex contains P and where W is the sum of the others. Owing t o Lemma 3.3(ii),
‘(1
= ‘NG(p)(‘)
Therefore, by Proposition 6.1(i), V ( P )2 U . It is plain that the homomorphism .Ic, corresponds to the map
En&p(V)
f
--$
EndF(U) efe
where e : VNG(p) --f U is the projection map, by identifying E n d ~ ( uwith )
It therefore follows that
Thus, by Proposition 6.l(iii), KeT.Ic, = K e r ( B $ ) for A = E n d ~ ( v ) . By the foregoing, we are left to verify that .Ic, is surjective. By Proposition 6.1(ii), we have U = I n v ( U p ) and so E n d ~ ( u = ) End,rp(U). Now fix X E E n d ~ ( uand ) denote by A‘ an element of E n d ~ p ( Vby ) extending X to V with 0 on W . Then it is obvious that .Ic, maps A‘ to X and the result is established. H