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Polynomial Ideals and Classes of Finite Groups
Journal of Algebra 229, 153᎐174 Ž2000. doi:10.1006rjabr.2000.8303, available online at http:rrwww.idealibrary.com on
Polynomial Ideals and Classes of Finite Groups Serge Bouc Uni¨ ersite´ Paris 7, 2 Place Jussieu, 75251 Paris Cedex 05, France Communicated by Michel Broue´ Received December 10, 1999
The object of this note is to discuss the properties of some polynomials Žon a countable set of indeterminates . attached to any finite group which generalize the Eulerian functions of a group defined by P. Hall Ž1936, Quart. J. Math. 7, 134᎐151.. In particular, I will define some classes of finite groups associated to prime ideals of the polynomial ring, and I will show that each finite group has a unique largest quotient in such a class of groups. This work is a generalization of the notion of the b-group introduced in an earlier paper ŽBouc, 1996, J. Algebra 183, 664᎐736. through a systematic use of the polynomial formalism of Section 7.2.5 there. For the reader’s convenience, however, this paper is self-contained and the proofs of the results already stated are included. 䊚 2000 Academic Press
1. THE POLYNOMIAL RING Ž1.1. Notation. I will consider the Žcountable. set S of isomorphism classes of finite Žnon-trivial. simple groups and the polynomial ring over S, R s ⺪ Ž X S . Sg S . In other words, the ring R is the algebra over ⺪ of the Grothendieck monoid of the category of finite groups. Ž1.2. Notation. If p is a prime number, and S is the isomorphism class of the cyclic group C p of order p, I denote by X p the variable X S of R. It is convenient to turn R into an ⺢-graded ring by setting deg Ž X S . s log < S < . If G is a finite group, then I define the monomial P Ž G . g R by P Ž G. s
Ł XS ŽG. , S
Sg S
153 0021-8693r00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.
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where for each finite simple group S the integer S Ž G . is the multiplicity of S as a composition factor of G. In particular P Ž1. s 1 and P Ž S . s X S if S g S . Note that with this definition, the degree of P Ž G . is equal to log < G < for any finite group G. I denote by P˜Ž G . the polynomial obtained from P Ž G . by Mobius ¨ inversion on the poset of subgroups of G, i.e., P˜Ž G . s
Ý Ž H , G. P Ž H . , HFG
where the notation H F G means that H is a subgroup of G and Ž H, G . is the Mobius function of the poset of subgroups of G. The monomial of ¨ highest degree in the expression of P˜Ž G . corresponds to the subgroup H s G, and the coefficient Ž G, G . is equal to 1. Thus deg Ž P˜Ž G . . s log < G < .
Ž 1.3.
Finally, if N is a normal subgroup of G, I define QG , N s
Ý
Ž H , G. P Ž H l N . .
HFG HNsG
This is a generalization of the previous formula since QG, G s P˜Ž G .. On the other hand, QG, 1 s 1 for any G. The terms of highest degree in QG, N correspond to subgroups H such that H l N s N and HN s G. Hence the only possible subgroup is H s G, and again the coefficient of P Ž H l N . s P Ž N . is equal to 1. Thus deg Ž QG , N . s log < N < .
Ž 1.4.
The polynomials P Ž G ., P˜Ž G ., and QG, N are invariant under group isomorphism: if : G ª G⬘ is an isomorphism, then obviously P Ž G . s P Ž G⬘., P˜Ž G . s P˜Ž G⬘., and QG, N s QG⬘, Ž N . . Ž1.5. Remark. The Eulerian function of the group G, defined by Hall w8x, is the function Ž G, s . defined for a complex number s by
Ž G, s . s
Ý Ž H , G. < H < s. HFG
Hence it is the evaluation of the polynomial P˜Ž G . when X S is replaced by < S < s, for all S g S . The name ‘‘Eulerian’’ comes from the fact that when s is a positive integer, the value Ž G, s . is the number of sequences of s elements of G which generate G. The polynomials P˜Ž G . and QG, N are also closely related to the probabilistic zeta function and its relative version, studied by K. Brown in w4x. The
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POLYNOMIAL IDEALS AND FINITE GROUPS
value Ž G, s . of this zeta function at a complex number s is given by 1r Ž G, s . s
Ž G, s . s
Ý HFG
Ž G, H . .
ŽBrown’s notation for 1r Ž G, s . is P Ž G, s ., but it is a bit confusing here and I prefer to use another symbol.. Hence the value Ž G, s .< G
Ý
Ž 1.8.
Ž H , G . P Ž H . s Ž R, G . P Ž RrN . QG , N
Ž 1.9.
Ž H , G . P Ž H l N . s Ž R, G . QG , N
Ž 1.10.
HFG HNsR
Ý HFG HNsR
P˜Ž G . s P˜Ž GrN . QG , N .
Ž 1.11.
Proof. The first equality is a trivial consequence of the definition, since the multiplicity of the simple group S as a composition factor of G is the sum of its multiplicities as a composition factor of N and GrN.
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For the second equality by the Crapo complementation formula Žsee w6, Theorems 3 and 5x or w5, pp. 420᎐421x., for H F G
Ž H , G. s
Ž H , K . Ž K , G. s
Ý HFKFG KNsG KlHNsH
Ž HN, G . Ž K , G . .
Ý HFKFG KNsG KlHNsH
Now if N F R F G, the conditions KN s G, K l HN s H, and HN s R are equivalent to KN s G and H s K l R. This gives
Ý
Ž H , G. P Ž H . s
HFG HNsR
Ý
Ž R, G . Ž K , G . P Ž K l R .
KFG KNsG
s Ž R, G .
Ž K , G. P Ž K l R. .
Ý KFG KNsG
Moreover, Ž K l R . N s R, thus RrN , Ž K l R .rŽ K l N . and P Ž K l R . s P Ž RrN . P Ž K l N . . It follows that for N F R F G
Ž H , G . P Ž H . s Ž R, G . P Ž RrN .
Ý HFG HNsR
Ý
Ž K , G. P Ž K l N .
KFG KNsG
and Eq. Ž1.9. follows. Equality Ž1.10. is a consequence, since if HN s R then P Ž H . s P Ž RrN . P Ž H l N .. Since, moreover, Ž R, G . s Ž RrN, GrN ., the summation of Ž1.9. for subgroups R with N F R F G gives P˜Ž G . s
Ý
Ž R, G . P Ž RrN . QG , N s P˜Ž GrN . QG , N ,
NFRFG
as was to be shown. Ž1.12. COROLLARY. If M and N are normal subgroups of G such that GrM , GrN, then QG, M s QG, N . Proof. Indeed, by Ž1.11. QG, N s P˜Ž G .rP˜Ž GrN .. 2. THE POLYNOMIALS P˜Ž G . In view of Ž1.11., it is natural to ask when the polynomial P˜ is irreducible.
POLYNOMIAL IDEALS AND FINITE GROUPS
Ž2.1. PROPOSITION. only if G is simple.
157
Let G be a finite group. Then P˜ is irreducible if and
Proof. Let S be a simple group. Then P˜Ž S . s X S q R where R is a polynomial in the variables X T , for simple groups T not isomorphic to S. Hence P˜Ž S . is irreducible. Conversely, if G is a finite group and P˜Ž G . is irreducible, let N be a normal subgroup of G. Then by Ž1.11. one of the polynomials P˜Ž GrN . or QG, N has degree 0. By Ž1.3. and Ž1.4., it follows that N s G or N s 1. Hence G is simple. Ž2.2. Notation. If N is a normal subgroup of G, denote by K G Ž N . the set of complements of N in G, i.e., K G Ž N . s L F G N LN s G, L l N s 1 4 . If M and N are normal subgroups of G, denote by K G Ž M, N . the set of subgroups of G which are complements of M and N, i.e., K G Ž M, N . s K G Ž M . l K G Ž N . . Ž2.3. PROPOSITION. Let G be a finite group and N be a minimal Ž non-tri¨ ial . abelian normal subgroup of G, isomorphic to Ž C p . n, for p prime and n ) 0. Then QG , N s X pn y < K G Ž N . < . Proof. Let H be a subgroup of G such that HN s G. Then H l N is normalized by H and by N since N is abelian. Hence either H l N s N and H s G, or H l N s 1 and H g K G Ž N .. In this case H is a maximal subgroup of G and Ž H, G . s y1. The proposition follows. Ž2.4. COROLLARY. Ž1.
Let G be a sol¨ able finite group and
1 s Ny1 - N0 - N1 - ⭈⭈⭈ - Nk s G be a chief series for G. Then for 0 F i F k, there exist a prime number pi and a positi¨ e integer n i such that NirNiy1 , Ž C p i . n i . Denote by m i the number of complements of NirNiy1 in the group GrNiy1. Then k
P˜Ž G . s
Ł Ž X pn y m i . . i
is0
i
Ž 2.5.
Ž2. Con¨ ersely, if G is a finite group, if there exists an integer k, and if there exist prime numbers pi and integers m i for 0 F i F k such that 2.5 holds, then G is sol¨ able.
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Proof. Assertion 1 follows from an obvious induction argument. For Assertion 2, suppose that G is a finite group such that Ž2.5. holds. Then the monomial of highest degree in P˜Ž G ., which is equal to P Ž G ., is the k product Ł is0 X pnii . This means that all the composition factors of G are cyclic, i.e., that G is solvable. Ž2.6. Remark. Corollary 2.4 can be viewed as a generalization of the Eulerian product formula obtained by Gaschutz ¨ w7x for the zeta function of a solvable group Žsee also w1x.. It should also be compared with the following question, cited by Brown ŽQuestion 1 of w4x.: QUESTION. If G is a finite group such that Ž G, s . has an Euler product ., is G sol¨ able? expansion with factors of the form 1rŽ1 y c i qys i 3. THE TWO NORMAL SUBGROUPS FORMULA Ž3.1. PROPOSITION w2, Lemme 20x. are normal subgroups of G then QG , N s
Ý
Let G be a finite group. If N and M
Ž L, G . P Ž L l M l N . QG r M , Ž L l N . M r M .
LFG LMsLNsG
Proof. By definition, QG , N s
Ž H , G. P Ž H l N . .
Ý HFG HNsG
As noted above
Ž H , G. s
Ž H , L . Ž L, G . .
Ý HFLFG LMsG LlHMsH
This gives QG , N s
Ý
Ý
Ž H , L . Ž L, G . P Ž H l N . .
HFG HFLFG HNsG LMsG LlHMsH
Now the conditions HN s G
HFL
LM s G
L l HM s H
are equivalent to the conditions LM s LN s G
HFL
HŽ L l N . s L
H G L l M.
POLYNOMIAL IDEALS AND FINITE GROUPS
159
It follows that QG , N s
Ž L, G .
Ý LFG LMsLNsG
Ý
Ž H , L. P Ž H l N . .
HFL HGLlM Ž H LlN .sL
The inner summation is equivalent to a summation over subgroups K s HrŽ L l M . of L s LrŽ L l M . such that K. J s L, where J denotes the normal subgroup Ž L l N .Ž L l M .rŽ L l M . of L. Moreover, since HN s G, P Ž H l N . s P Ž H . P Ž N . rP Ž G . s P Ž K . P Ž N . P Ž L l M . rP Ž G . s s s
P Ž K l J . P Ž L. PŽ J .
P Ž N . P Ž L l M . rP Ž G .
P Ž K l J . P Ž L. P Ž N . P Ž J . P Ž G. PŽ K l J . PŽ L l N . P Ž L l NrL l N l M .
s PŽ K l J . PŽ L l M l N .. Thus QG , N s
Ý
Ž L, G . P Ž L l M l N .
LFG LMsLNsG
s
Ý
Ý
Ž K , L. P Ž K l J .
KFL KJsL
Ž L, G . P Ž L l M l N . Q L , J .
LFG LMsLNsG
The formula follows since L , GrM and since the image of J under this isomorphism is Ž L l N . MrM. Ž3.2. COROLLARY.
If M F N, then QG , N s QG , M QG r M , N r M .
Proof. This follows from the fact that if LM s G and M F N, then Ž L l N . M s N. This corollary also has an obvious direct proof using 1.11.
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4. I-GROUPS One of the methods used in w2x was to replace each variable X S by < S < and to look whether the resulting number QG, N is zero. This can be generalized by considering an ideal I of R and looking whether QG, N is in I or not. Ž4.1. CONVENTION. In the following, the expression ‘‘the group H is a quotient of the group G’’ means that H is isomorphic to a factor group of G. Ž4.2. DEFINITION. Let I be an ideal of R. A finite group G is called an I-group if for any non-trivial normal subgroup N of G, the polynomial QG, N belongs to I . Ž4.3. PROPOSITION. Let G be a finite group and I an ideal of R. If M and N are normal subgroups of G and GrM is an I-group, then QG , N '
Ý
Ž L, G . P Ž L l N .
Ž mod I . .
LFG LMsLNsG LlNFLlM
In particular, if QG, N f I , then GrM is a quotient of GrN. Proof. The first assertion follows from Proposition 3.1 and from the definition of an I-group. Now if QG, N f I , then there exists a subgroup L of G such that LM s LN s G and L l N F L l M. Now GrM , LrŽ L l M . is a quotient of GrN , LrŽ L l N .. Ž4.4. PROPOSITION. Let G be a finite group and I be a prime ideal of R. There exists a factor group  I Ž G . of G characterized uniquely up to isomorphism by the following properties: 1. The group  I Ž G . is an I-group. 2. If K is a quotient of G and K is an I-group, then K is a quotient of  I Ž G .. Moreo¨ er, if N is a normal subgroup of G, then the following conditions are equi¨ alent: Ža. The group  I Ž G . is a quotient of GrN. Žb.  I Ž GrN . ,  I Ž G .. Žc. QG r N f I . Proof. Properties 1 and 2 clearly show that the group  I Ž G . is unique up to isomorphism, if it exists. Let M be a normal subgroup of G, maximal subject to QG, M f I . Then by Corollary 3.2, if N is a normal subgroup of G strictly containing M, QG , N s QG , M QG r M , N r M g I .
POLYNOMIAL IDEALS AND FINITE GROUPS
161
Since QG, M f I and since I is prime, it follows that QG r M , N r M g I . This holds for any non-trivial normal subgroup NrM of GrM, hence GrM is an I-group. By Proposition 4.3, since QG, M f I , it follows that any I-group which is a quotient of G is a quotient of GrM. Thus  I Ž G . s GrM has Properties 1 and 2. Note that by construction and by Ž1.12., if M is a normal subgroup of G such that GrM ,  I Ž G ., then QG, M f I . Now if N is a normal subgroup of G, then the group  I Ž GrN . is an I-group, which is quotient of GrN, and hence of G. Thus  I Ž GrN . is always a quotient of  I Ž G .. If Ža. holds, then  I Ž G . is an I-group, which is a quotient of GrN. Hence  I Ž G . is a quotient of  I Ž GrN ., and Žb. holds. If Žb. holds, and if MrN is a normal subgroup of GrN such that
Ž GrN . r Ž MrN . ,  I Ž GrN . , then GrM ,  I Ž GrN . ,  I Ž G .. Hence as noted above QG, M f I , and since QG, M is a multiple of QG, N , it follows that QG, N f I . Hence Žc. holds. If Žc. holds, then by Proposition 4.3 the group  I Ž G . is a quotient of GrN and Ža. holds. There are generally several normal subgroups M of G such that GrM ,  I Ž G .. They are related as follows: Ž4.5. PROPOSITION. Let G be a finite group and I be a prime ideal of R. Let M and N be normal subgroups of G such that GrM , GrN ,  I Ž G . . Let x M l N, M w G denote the poset of normal subgroups of G which contain strictly M l N and are strictly contained in M, and let n G Ž M, N . denote its reduced Euler᎐Poincare´ characteristic Ž with the con¨ ention n G Ž M, N . s 1 if M s N .. Then: 1. There exists an automorphism of the group GrM l N such that
Ž MrM l N . s NrM l N. Hence the posets x M l N, M w G and x M l N, N w G are isomorphic and in particular n G Ž M, N . s n G Ž N, M .. 2. The group MrM l N , NrM l N is isomorphic to a direct product of simple groups. 3. QG, N ' QG, M l N n G Ž M, N .< K G r M l N Ž MrM l N, NrM l N .< Žmod I ..
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Proof. Since QG, N s QG, M l N QG r M l N, N r M l N by Corollary 3.2, replacing G by GrM l N shows that it suffices to consider the case M l N s 1. In this case the groups M and N centralize each other. By Proposition 4.3, since moreover GrM , GrN implies L l M s L l N whenever LM s LN s G and L l N F L l M, QG , N '
Ž L, G .
Ý
Ž mod I . .
LFG LMsLNsG LlMsLlNs1
In other words, QG , N '
Ý
Ž L, G .
Ž mod I . .
LgK GŽ M , N .
Since QG, N f I , this shows in particular that K G Ž M, N . / ⭋. Now fix L g K G Ž M, N . and define a map from M to N by Ž m. s n if mn g L. This is well defined since L g K G Ž M, N .. Moreover, if m and m⬘ are in M, n s Ž m., and n⬘ s Ž m⬘., then mnm⬘n⬘ s mm⬘nn⬘ g L since M and N commute. This shows that Ž mm⬘. s nn⬘, hence is a group homomorphism from M to N. By symmetry, the map from N to M is defined by Ž n. s m if nm g L is also a group homomorphism, which is clearly the inverse of , since M and N commute. Moreover, the maps and are clearly L equivariant. Now define a map : G ª G by
Ž ml . s Ž m . l
᭙ m g M, ᭙l g L.
Then for m, m⬘ in M and l, l⬘ in L,
Ž mlm⬘l⬘ . s Ž m.l m⬘.l l⬘ . s Ž m . .l Ž m⬘ . .l l⬘ s Ž m . l Ž m⬘ . l⬘ s Ž ml . Ž m⬘l⬘ . . This shows that is a group homomorphism, which is clearly an automorphism of G. By construction Ž M . s N, thus induces an isomorphism of posets from x1, M w G to x1, N w G , and n G Ž M, N . s n G Ž N, M .. This shows Assertion 1. Now the maps X g w 1, N x ¬ LX g w L, G x L
Y g w L, G x ¬ Y l N g w 1, N x
L
are inverse isomorphisms of posets. Moreover, w1, N x L s w1, N x L M s w1, N xG since M and N commute. Thus Ž L, G . is equal to the reduced
POLYNOMIAL IDEALS AND FINITE GROUPS
163
Euler᎐Poincare ´ characteristic of the poset x1, N w G s w1, N xG y 1, N 4, and this does not depend on the choice of L in K G Ž M, N .. Thus QG , N ' ˜ Ž x 1, N w
G
. < K G Ž M, N . <
Ž mod I .
and this proves Assertion 3. Since QG, N f I , it follows from the Crapo complementation formula that every normal subgroup of G contained in N has a complement in N, invariant by G. This can only happen if N is a direct product of simple groups, and this completes the proof of the proposition. Ž4.6. EXAMPLES. The case of b-groups discussed in w2x corresponds to the ideal I generated by all X S y < S <, for S g S . In this case if G is a p-group then  I Ž G . is trivial if G is cyclic, and is isomorphic to Ž C p . 2 otherwise. This follows from Proposition 14 of w2x, but I will give another more general proof here in Corollary 7.4. If ⌽ Ž G . is the Frattini subgroup of G, and if the order of Gr⌽Ž G . is at least p 3, then there are several normal subgroups N of G such that GrN , Ž C p . 2 , but all those subgroups contain ⌽ Ž G . and are conjugate by the automorphism group of Gr⌽Ž G ..
5. EXAMPLE: THE IDEAL OF VALUATION Ž5.1. Notation. I denote by V the ideal of R generated by all X S , for S g S. In other words, the ideal V is the ideal of polynomials with constant V is isomorphic to ⺪, term equal to zero. Clearly the quotient ring RrV thus V is a prime ideal. By definition, if N is a normal subgroup of the finite group G, then QG , N s
Ý
Ž L, G . P Ž L l N . .
L-G LNsG
In this sum, the monomial P Ž L l N . is in V , unless L l N s 1. It follows that QG , N '
Ý
Ž L, G .
Ž mod V . .
LgK GŽ N .
If N is a minimal normal subgroup of G and if N is abelian, then by Proposition 2.3 QG , N ' y< K G Ž N . <
Ž mod V .
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In particular, QG, N g V if and only if K G Ž N . s ⭋. This is equivalent to requiring N to be contained in each maximal subgroup L of G, i.e., N to be contained in ⌽ Ž G .. Ž5.2. Notation. If G is a finite group, denote by M Ž G . the subgroup generated by all minimal normal subgroups of G. Ž5.3. PROPOSITION. Let G be a sol¨ able finite group. Then G is a V-group if and only if M Ž G . F ⌽ Ž G .. Proof. This follows clearly from the previous discussion. Ž5.4. Remark. If G is an arbitrary finite group and if M Ž G . F ⌽ Ž G ., then G is a V-group, but the converse is false in general: for example, a simple group S is a V-group if and only if Ž1, S . s 0. This happens for instance if S is the simple group of order 168. Still, M Ž S . s S is not contained in ⌽ Ž S . s 1. This remark and Proposition 4.4 suggest the following variation: Ž5.5. Notation. Denote by A the ideal of R generated by all X p for p prime. A is isomorphic to the ring ⺪wŽ X S .S g S 0 x of polynomials on Clearly RrA the set S 0 of isomorphism classes of non-abelian simple groups. Hence A is a prime ideal of R. Ž5.6. PROPOSITION. Let G be a finite group. Then G is an A-group if and only if M Ž G . F ⌽ Ž G .. Proof. If M Ž G . F ⌽ Ž G ., since ⌽ Ž G . is nilpotent all minimal normal subgroups of G are abelian and have no complement in G. If N is a minimal normal subgroup of G isomorphic to Ž C p . n, for a prime number p and a positive integer n, then by Proposition 2.3 QG , N s X pn y < K G Ž N . < s X pn g A . Hence G is an A-group. Conversely, if G is an A-group and N is a minimal normal abelian subgroup of G, isomorphic to Ž C p . n, then QG , N s X pn y < K G Ž N . < ' y< K G Ž N . < Ž mod A . Thus QG, N g A if and only if K G Ž N . s ⭋, or equivalently since N is abelian, if N F ⌽ Ž G .. Now suppose N is a non-abelian minimal normal subgroup of G. Then N , S n, for some non-abelian simple group S and some positive integer n. But QG , N s Ý Ž H , G . P Ž H l N . HFG HNsG
POLYNOMIAL IDEALS AND FINITE GROUPS
165
and the term of highest degree in this expression is obtained for H s G, and it is equal to P Ž N . s X Sn. But the ideal A consists of linear combinations of monomials Ł S g S X S␣ S for which the exponent ␣ S is positive at least for one abelian simple group S Ži.e., the monomials in which some variable X p for p prime appears.. This shows that QG, N cannot belong to A if N is non-abelian. Hence all the minimal normal subgroups of G are abelian, and it follows that M Ž G . F ⌽ Ž G ., as was to be shown. Ž5.7. COROLLARY. Let G be a finite group. Then there exists a factor group H of G, characterized uniquely up to isomorphism by the following properties: 1. M Ž H . F ⌽ Ž H .. 2. If K s GrN is a quotient of G such that M Ž K . F ⌽ Ž K ., and if M1 ᎐ G is such that GrM , H, then there exists a subgroup L of G with LM s LN s G
L l M F L l N,
and in particular K is a quotient of H. Proof. Of course H s  A Ž G .. Ž5.8. EXAMPLE. When G is a p-group, for some prime p, it is possible to describe explicitly the quotient  V Ž G . s  A Ž G .. Indeed, this quotient is obtained by considering a normal subgroup N of G, maximal subject to QG, N f V . With the notation of Proposition 2.4, if 1 s Ny1 - N0 - ⭈⭈⭈ - Nk s G is a chief series for G, and if N s Nl , for y1 F l F k, then l
QG , N s P˜Ž G . rP˜Ž GrN . s ' Ž y1 .
lq1
Ł Ž X pn y m i . i
is0
l
Ł mi
Ž mod V . .
is0
Thus N is a maximal normal subgroup such that all the m i ’s are non-zero. In particular, every minimal normal subgroup of G contained in N must have a complement in G, and hence in N. It follows that N is elementary abelian and G-semi-simple. Hence N is central. Moreover, if L irNiy1 g K G r N iy 1Ž NirNiy1 ., for 0 F i F l, then it is easy to see that L0 l L1 l ⭈⭈⭈ l L l is a complement of Nl s N in G. Thus G can be split as a direct product, G s N = L.
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Moreover, L is a V-group, thus M Ž L. F ⌽ Ž L.. Since L is a p-group, all the minimal normal subgroups of L are central of order p. It follows that no central subgroup of order p can have a complement in L. Hence N is a maximal elementary abelian central subgroup of G having a complement in G and  V Ž G . is the quotient of G by its ‘‘largest elementary abelian direct summand.’’ In this case, one can say exactly how many subgroups N such that GrN ,  V Ž G . there are: indeed, with the previous notations, the group M Ž L. is equal to the subgroup ⍀ 1Ž ZŽ L.. generated by the central elements of order p of L. But ⌽ Ž G. s 1 = ⌽ Ž L. and ⍀ 1Ž Z Ž G . . s N = ⍀ 1Ž Z Ž L . . .
Ž 5.9.
Taking the intersection of those two equations gives ⌽ Ž G . l ⍀ 1Ž Z Ž G . . s 1 = ⍀ 1Ž Z Ž L . . . It follows from 5.9 that N must be a complement of ⌽ Ž G . l ⍀ 1Ž ZŽ G .. in ⍀ 1Ž ZŽ G ... Conversely, if N is such a complement, there exists a subgroup K of G, containing ⌽ Ž G ., such that Kr⌽Ž G . is a complement of N⌽ Ž G .r⌽ Ž G . in Gr⌽Ž G ., since Gr⌽Ž G . is elementary abelian. In other words, KN s G
K l N⌽ Ž G . s ⌽ Ž G . .
It follows that K l N F N l ⌽ Ž G . s 1, thus K is a complement of N. Clearly now N is G-semi-simple, and it follows that QG, N f V . Thus the subgroups N such that GrN ,  I Ž G . are exactly the complements of ⌽ Ž G . l ⍀ 1Ž ZŽ G .. in ⍀ 1Ž ZŽ G ... 6. DIRECT PRODUCTS Ž6.1. Notation. Let G and H be finite groups. Denote by p1 and p 2 the projections from G = H to G and H respectively. If L is a subgroup of G = H, set p1 Ž L . s g g G N ᭚h g H , Ž g , h . g L4 k 1 Ž L . s g g G N Ž g , 1 . g L4 p 2 Ž L . s h g H N ᭚ g g G, Ž g , h . g L4 k 2 Ž L . s h g H N Ž 1, h . g L4 .
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167
Then k i Ž L. 1 ᎐ pi Ž L. for i s 1, 2, and the groups q Ž L. s LrŽ k 1Ž L. k 2 Ž L.., Ž . Ž . p1 L rk 1 L , and p 2 Ž L.rk 2 Ž L. are canonically isomorphic. Let G and H be finite groups. Then
Ž6.2. PROPOSITION.
P˜Ž G . P˜Ž H . s
P˜Ž L . .
Ý LFG=H p 1Ž L .sG p 2 Ž L .sH
Proof. This is a consequence of the definition of the polynomials P˜ by Mobius inversion. Indeed, for any finite groups G and H ¨ PŽG = H . s
P˜Ž L . .
Ý
Ž 6.3.
LFG=H
Now setting
Ž G, H . s
Ý
P˜Ž L . ,
LFG=H p 1Ž L .sG p 2 Ž L .sH
the right-hand side of Eq. Ž6.3. can be written as
Ž A, B . .
Ý AFG BFH
Thus
Ý Ž C, G . Ž D, H . P Ž C = D . CFG DFH
Ý Ý Ž C, G . Ž D, H . Ž A, B .
s
CFG AFC DFH BFD
s
Ý AFG BFH
ž
Ý
Ž C, G . Ž D, H . Ž A, B .
AFCFG BFDFH
/
s Ž G, H . . The proposition follows, since the left-hand side is equal to P˜Ž G . P˜Ž H ., because P Ž C = D . s P Ž C . P Ž D . for any C F G and D F H. Ž6.4. COROLLARY. group, then
If G and H ha¨ e no non-tri¨ ial isomorphic factor P˜Ž G = H . s P˜Ž G . P˜Ž H . .
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Proof. Indeed, in this case the only subgroup L of G = H such that p1Ž L. s G and p 2 Ž L. s H is G = H itself, since q Ž L. is a quotient of both G and H, hence it is trivial. Ž6.5. PROPOSITION w2, Lemme 22x.
P˜Ž G = H . s P˜Ž G . P˜Ž H .
Let G and H be finite groups. Then
Ý LFG=H p 1Ž L .sG p 2 Ž L .sH
˜ Ž x 1, q Ž L . w
qŽ L.
P˜Ž q Ž L . .
..
Proof. Denote by K the group G = H and by M and N the normal subgroups G = 1 and 1 = H of K. Then by Proposition 3.1 since M l N s1 QK , N s
Ž L, K . Q K r M , Ž L l N . M r M .
Ý LFK LMsLNsK
The condition LM s LN s K is equivalent to p1Ž L. s G and p 2 Ž L. s H. In this case, moreover, the maps X g x L, K w ¬ k 1 Ž X . rk 1 Ž L . g x 1, Grk 1 Ž L . w Yrk 1 Ž L . g x 1, Grk 1 Ž L . w
G
G
¬ Ž g , h . g G = H N ᭚a g G, Ž a, h . g L, gay1 g Y 4 are mutual inverse isomorphisms of posets. Since Grk 1Ž L. , q Ž L., it follows that Ž L, G . s ˜ Žx1, q Ž L.w qŽ L. .. Moreover, Q K r M , Ž L l N . M r M s Q L r L l M , Ž L l N .Ž L l M .r L l M s
P˜Ž LrL l M . P˜Ž Lr Ž L l N . Ž L l M . .
.
But LrL l M , KrM , H and Lr Ž L l N . Ž L l M . s Lr Ž k 1 Ž L . = k 2 Ž L . . s q Ž L . .
POLYNOMIAL IDEALS AND FINITE GROUPS
169
It follows that
Q K , N s P˜Ž H .
Ý LFG=H p 1Ž L .sG p 2 Ž L .sH
˜ Ž x 1, q Ž L . w
qŽ L.
P˜Ž q Ž L . .
..
The proposition follows, since Q K , N s P˜Ž K .rP˜Ž KrN . s P˜Ž K .rP˜Ž G .. Ž6.7. COROLLARY. Let G and H be finite groups, and denote by Gs , Hs , and Ž G = H .s , Gs = Hs the respecti¨ e largest semi-simple quotients of G, H, and G = H. Then P˜Ž G = H . P˜Ž Ž G = H . s .
s
P˜Ž G . P˜Ž H . P˜Ž Gs . P˜Ž Hs .
.
Proof. The only non-zero terms in the formula Ž6.6. correspond to subgroups L of G = H such that q Ž L. is semi-simple. Moreover, since p1Ž L. s G and p 2 Ž L. s H, the group q Ž L. is a quotient of G and H. It follows that if M and N are normal subgroups of G and H respectively, such that GrM , Gs and HrN , Hs , then L G M = N. Then the group L⬘ s LrM = N is a subgroup of Gs = Hs , Ž G = H .s and q Ž L . s Lr Ž k 1 Ž L . = k 2 Ž L . . , Ž LrM = N . r Ž k 1 Ž L . = k 2 Ž L . rM = N . , L⬘r Ž Ž k 1 Ž L . rM . = Ž k 2 Ž L . rN . . s q Ž L⬘ . . The corollary follows, since the correspondence L ¬ L⬘ from
K F G = H N p1Ž K . s G, p2 Ž K . s H , K G M = N 4 to the set
K ⬘ F Gs = Hs N p1Ž K ⬘. s Gs , p2 Ž K ⬘. s Hs 4 is clearly one to one. Corollary 6.7 gives a way to compute P˜Ž G = H . knowing P˜Ž G ., P˜Ž H ., and the groups Gs and Hs . The following notation will be convenient: Ž6.8. Notation. If G and H are finite groups, denote by sŽ G, H . the number of surjective group homomorphisms from G to H.
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SERGE BOUC
If n, m g N, set ny1
F Ž G, n . s
Ł Ž P˜Ž G . y s Ž G i , G . .
is0
B Ž G, m, n . s
F Ž G, m q n . F Ž G, m . F Ž G, n .
.
Here the letter F stands for ‘‘factorial’’ and the letter B for ‘‘binomial.’’ Note that B Ž G, m, n. is an element of the field of fractions of R. Ž6.9. PROPOSITION. Ž1. If S is a finite simple group and n g N, then P˜Ž S n . s F Ž S, n.. Moreo¨ er, sŽ S n, S . is equal to p n y 1 if S , C p , and to n
Ł B Ž S, aS , bS . .
Sg S
Proof. The first formula follows from an easy induction argument, using Proposition 6.2 or Ž6.6.. Using this, the second formula follows from Corollaries 6.4 and 6.7. Ž6.10. LEMMA. Let I be a prime ideal of R. If G and H are I-groups, and if they ha¨ e no non-tri¨ ial isomorphic factor group, then G = H is an I-group. Proof. Since G and H are quotients of G = H, it follows that they are quotients of  I Ž G = H .. Hence there is a group homomorphism
: I Ž G = H . ª G = H such that p1 ( and p 2 ( are surjective. Let K denote the image of . Then p1Ž K . s G and p 2 Ž K . s H. Thus q Ž K . is a quotient of both G and H, hence it is trivial. It follows that K s G = H, hence  I Ž G = H . , G = H. Thus G = H is an I-group. Ž6.11. PROPOSITION. Let I be a prime ideal of R, and G and H be finite groups ha¨ ing no non-tri¨ ial isomorphic factor group. 1.
If M 1 ᎐ G and N 1 ᎐ H, then QG= H , M=N s QG , M Q H , N .
2.
Moreo¨ er,
I Ž G = H . , I Ž G. = I Ž H . .
POLYNOMIAL IDEALS AND FINITE GROUPS
171
3. The group G = H is an I-group if and only if G and H are I-groups. Proof. Let M be a normal subgroup of G and N be a normal subgroup of H. Then by Corollary 6.4, since GrM and HrN have no non-trivial isomorphic factor group, P˜Ž G = H . s P˜Ž G . P˜Ž H . P˜Ž G = HrM = N . s P˜Ž GrM = HrN . s P˜Ž GrM . P˜Ž HrN . . Taking the quotient of those equations gives Assertion 1, QG= H , M=N s QG , M Q H , N . Clearly  I Ž G . and  I Ž H . have no common non-trivial factor group. Hence by Lemma 6.10  I Ž G . =  I Ž H . is an I-group, and it is a quotient of G = H. Hence it is a quotient of  I Ž G = H .. Conversely, if M 1 ᎐ G is such that GrM ,  I Ž G . and N 1 ᎐ H is such that HrN ,  I Ž H ., then QG, M f I and Q H , N f I , thus QG=H , M=N f I by Assertion 1 since I is prime. By Proposition 4.3 the group  I Ž G = H . is a quotient of
Ž G = H . rŽ M = N . ,  I Ž G. =  I Ž H . . Assertion 2 follows. Finally, G = H is an I-group if and only if G = H s  I Ž G = H .. By Assertion 2, this is equivalent to G s  I Ž G . and H s  I Ž H ., which proves Assertion 3.
7. NILPOTENT I-GROUPS It is possible to describe all the nilpotent I-groups when I is a prime ideal of R. Since any nilpotent group is isomorphic to the direct product of its Sylow p-subgroups, for prime numbers p, it follows from Proposition 6.11 that a nilpotent group is an I-group if and only if all its Sylow subgroups are I-groups. Thus in order to describe the nilpotent I-groups, it suffices to describe the p-groups which are also I-groups. Ž7.1. PROPOSITION. Let I be a prime ideal of R, let p be a prime number, and let G be a finite p-group. Then 1. If G is elementary abelian of order p n, then G is an I-group if and only if n s 0 or n G 1 and X p y p ny 1 g I .
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SERGE BOUC
2. If G is not elementary abelian, then G is an I-group if and only if X p g I and one of the following holds: Ža. p g I Žb. ⍀ 1Ž ZŽ G .. F ⌽ Ž G ., or equi¨ alently, the group G cannot be written G , C p = H, for some finite group H. Proof. Suppose G is non-trivial and let N be a minimal normal subgroup of G. Then N is central of order p and QG , N s X p y < K G Ž N . < , since any proper subgroup L of G such that LN s G is a complement of G, and L is a maximal subgroup of G in this case. Suppose that G is elementary abelian of order p n. If n s 0, then G is trivial and G is an I-group. If n G 1 and if N is a subgroup of G of order p, then QG , N s X p y p ny 1 since there are p ny 1 complements of N in G. This proves Assertion 1. If G is not elementary abelian, then there exists a central subgroup N of G of order p contained in ⌽ Ž G .. If N is such a subgroup, then K G Ž N . s ⭋ and QG , N s X p .
Ž 7.2.
Now if M is a central subgroup of G of order p and M g ⌽ Ž G ., then QG , N s X p y < K G Ž M . < .
Ž 7.3.
Moreover, in this case K G Ž M . / ⭋, thus there exists a group H such that G , M = H and < K G Ž M . < s
POLYNOMIAL IDEALS AND FINITE GROUPS
173
Proposition 7.1 can be reformulated as follows: Ž7.4. COROLLARY. Let I be a prime ideal of R, let p be a prime number, and let G be a finite p-group. Then: If X p g I and p g I , then G is an I-group if and only if G ` C p . If X p g I and p f I , then G is an I-group if and only if ⍀ 1Ž ZŽ G .. F ⌽ Ž G .. If X p f I , then let h, k g ⺞ j ⬁4 defined by 䢇 䢇
䢇
h s Inf l g ⺞ N X p y p l g I 4 k s Inf l g ⺞ y 0 4 N 1 y p l g I 4 . Then G is an I-group if and only if G is elementary abelian of order p n with n s 0, n ' h q 1 Ž k . if h / ⬁ and k / ⬁, or n s h q 1 if h / ⬁ and k s ⬁. Proof. Indeed, if X p g I and p g I , then by Proposition 7.1 the only case where G is not an I-group is the case G , C p . If X p g I and if a non-trivial elementary abelian p-group of order p n is an I-group, then p ny 1 g I , hence 1 g I or p g I . Thus if p f I , then no non-trivial elementary abelian p-group can be an I-group, and G is an I-group if and only if ⍀ 1Ž ZŽ G .. F ⌽ Ž G .. Finally, if X p f I , then G is an I-group if and only if G is elementary abelian of order p n with n s 0 or n G 1 and X p y p ny 1 g I . This cannot happen if h s ⬁, and if h g ⺞ this is equivalent to p h y p ny 1 s p h Ž 1 y p ny hy1 . g I . But p h f I , since otherwise X p s Ž X p y p h . q p h g I , and this is equivalent to 1 y p ny hy1 g I , or n ' h q 1Ž k ., which has to be viewed as an equality if k s ⬁. Ž7.5. Remark. The last case of Corollary 7.4 should be compared with Theorem 8.2 of w3x, which deals with ‘‘b-groups in characteristic q,’’ i.e., with the case where I is generated by a prime number q / p Žor q s 0., and by X S y < S <, for all simple groups S. REFERENCES 1. S. Bouc, Homologie de certains ensembles ordonnes, ´ C. R. Acad. Sci. Paris Ser. I 299, No. 2 Ž1984., 9᎐12. 2. S. Bouc, Foncteurs d’ensembles munis d’une double action, J. Algebra 183 Ž1996., 664᎐736; doi: 10116.jabr.1996.0236.
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3. S. Bouc and J. Thevenaz, The group of endo-permutation modules, In¨ ent. Math. 139 ´ Ž2000., 275᎐349. 4. K. Brown, The coset poset and the probabilistic zeta function of a finite group, preprint, 1999. 5. K. Brown and J. Thevenaz, A generalization of Sylow’s third theorem, J. Algebra 115 ´ Ž1988., 414᎐430. 6. H. Crapo, The Mobius function of a lattice, J. Combin. Theory 1 Ž1966., 126᎐131. ¨ 7. W. Gaschutz, Gruppen, Illinois J. Math. 3 ¨ Die Eulersche Funktion endlicher auflosbarer ¨ Ž1959., 469᎐476. 8. P. Hall, The Eulerian functions of a group, Quart. J. Math. 7 Ž1936., 134᎐151.
Polynomial Ideals and Classes of Finite Groups Serge Bouc Uni¨ ersite´ Paris 7, 2 Place Jussieu, 75251 Paris Cedex 05, France Communicated by Michel Broue´ Received December 10, 1999
The object of this note is to discuss the properties of some polynomials Žon a countable set of indeterminates . attached to any finite group which generalize the Eulerian functions of a group defined by P. Hall Ž1936, Quart. J. Math. 7, 134᎐151.. In particular, I will define some classes of finite groups associated to prime ideals of the polynomial ring, and I will show that each finite group has a unique largest quotient in such a class of groups. This work is a generalization of the notion of the b-group introduced in an earlier paper ŽBouc, 1996, J. Algebra 183, 664᎐736. through a systematic use of the polynomial formalism of Section 7.2.5 there. For the reader’s convenience, however, this paper is self-contained and the proofs of the results already stated are included. 䊚 2000 Academic Press
1. THE POLYNOMIAL RING Ž1.1. Notation. I will consider the Žcountable. set S of isomorphism classes of finite Žnon-trivial. simple groups and the polynomial ring over S, R s ⺪ Ž X S . Sg S . In other words, the ring R is the algebra over ⺪ of the Grothendieck monoid of the category of finite groups. Ž1.2. Notation. If p is a prime number, and S is the isomorphism class of the cyclic group C p of order p, I denote by X p the variable X S of R. It is convenient to turn R into an ⺢-graded ring by setting deg Ž X S . s log < S < . If G is a finite group, then I define the monomial P Ž G . g R by P Ž G. s
Ł XS ŽG. , S
Sg S
153 0021-8693r00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.
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SERGE BOUC
where for each finite simple group S the integer S Ž G . is the multiplicity of S as a composition factor of G. In particular P Ž1. s 1 and P Ž S . s X S if S g S . Note that with this definition, the degree of P Ž G . is equal to log < G < for any finite group G. I denote by P˜Ž G . the polynomial obtained from P Ž G . by Mobius ¨ inversion on the poset of subgroups of G, i.e., P˜Ž G . s
Ý Ž H , G. P Ž H . , HFG
where the notation H F G means that H is a subgroup of G and Ž H, G . is the Mobius function of the poset of subgroups of G. The monomial of ¨ highest degree in the expression of P˜Ž G . corresponds to the subgroup H s G, and the coefficient Ž G, G . is equal to 1. Thus deg Ž P˜Ž G . . s log < G < .
Ž 1.3.
Finally, if N is a normal subgroup of G, I define QG , N s
Ý
Ž H , G. P Ž H l N . .
HFG HNsG
This is a generalization of the previous formula since QG, G s P˜Ž G .. On the other hand, QG, 1 s 1 for any G. The terms of highest degree in QG, N correspond to subgroups H such that H l N s N and HN s G. Hence the only possible subgroup is H s G, and again the coefficient of P Ž H l N . s P Ž N . is equal to 1. Thus deg Ž QG , N . s log < N < .
Ž 1.4.
The polynomials P Ž G ., P˜Ž G ., and QG, N are invariant under group isomorphism: if : G ª G⬘ is an isomorphism, then obviously P Ž G . s P Ž G⬘., P˜Ž G . s P˜Ž G⬘., and QG, N s QG⬘, Ž N . . Ž1.5. Remark. The Eulerian function of the group G, defined by Hall w8x, is the function Ž G, s . defined for a complex number s by
Ž G, s . s
Ý Ž H , G. < H < s. HFG
Hence it is the evaluation of the polynomial P˜Ž G . when X S is replaced by < S < s, for all S g S . The name ‘‘Eulerian’’ comes from the fact that when s is a positive integer, the value Ž G, s . is the number of sequences of s elements of G which generate G. The polynomials P˜Ž G . and QG, N are also closely related to the probabilistic zeta function and its relative version, studied by K. Brown in w4x. The
155
POLYNOMIAL IDEALS AND FINITE GROUPS
value Ž G, s . of this zeta function at a complex number s is given by 1r Ž G, s . s
Ž G, s . s
Ý HFG
Ž G, H . .
ŽBrown’s notation for 1r Ž G, s . is P Ž G, s ., but it is a bit confusing here and I prefer to use another symbol.. Hence the value Ž G, s .< G
Ý
Ž 1.8.
Ž H , G . P Ž H . s Ž R, G . P Ž RrN . QG , N
Ž 1.9.
Ž H , G . P Ž H l N . s Ž R, G . QG , N
Ž 1.10.
HFG HNsR
Ý HFG HNsR
P˜Ž G . s P˜Ž GrN . QG , N .
Ž 1.11.
Proof. The first equality is a trivial consequence of the definition, since the multiplicity of the simple group S as a composition factor of G is the sum of its multiplicities as a composition factor of N and GrN.
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For the second equality by the Crapo complementation formula Žsee w6, Theorems 3 and 5x or w5, pp. 420᎐421x., for H F G
Ž H , G. s
Ž H , K . Ž K , G. s
Ý HFKFG KNsG KlHNsH
Ž HN, G . Ž K , G . .
Ý HFKFG KNsG KlHNsH
Now if N F R F G, the conditions KN s G, K l HN s H, and HN s R are equivalent to KN s G and H s K l R. This gives
Ý
Ž H , G. P Ž H . s
HFG HNsR
Ý
Ž R, G . Ž K , G . P Ž K l R .
KFG KNsG
s Ž R, G .
Ž K , G. P Ž K l R. .
Ý KFG KNsG
Moreover, Ž K l R . N s R, thus RrN , Ž K l R .rŽ K l N . and P Ž K l R . s P Ž RrN . P Ž K l N . . It follows that for N F R F G
Ž H , G . P Ž H . s Ž R, G . P Ž RrN .
Ý HFG HNsR
Ý
Ž K , G. P Ž K l N .
KFG KNsG
and Eq. Ž1.9. follows. Equality Ž1.10. is a consequence, since if HN s R then P Ž H . s P Ž RrN . P Ž H l N .. Since, moreover, Ž R, G . s Ž RrN, GrN ., the summation of Ž1.9. for subgroups R with N F R F G gives P˜Ž G . s
Ý
Ž R, G . P Ž RrN . QG , N s P˜Ž GrN . QG , N ,
NFRFG
as was to be shown. Ž1.12. COROLLARY. If M and N are normal subgroups of G such that GrM , GrN, then QG, M s QG, N . Proof. Indeed, by Ž1.11. QG, N s P˜Ž G .rP˜Ž GrN .. 2. THE POLYNOMIALS P˜Ž G . In view of Ž1.11., it is natural to ask when the polynomial P˜ is irreducible.
POLYNOMIAL IDEALS AND FINITE GROUPS
Ž2.1. PROPOSITION. only if G is simple.
157
Let G be a finite group. Then P˜ is irreducible if and
Proof. Let S be a simple group. Then P˜Ž S . s X S q R where R is a polynomial in the variables X T , for simple groups T not isomorphic to S. Hence P˜Ž S . is irreducible. Conversely, if G is a finite group and P˜Ž G . is irreducible, let N be a normal subgroup of G. Then by Ž1.11. one of the polynomials P˜Ž GrN . or QG, N has degree 0. By Ž1.3. and Ž1.4., it follows that N s G or N s 1. Hence G is simple. Ž2.2. Notation. If N is a normal subgroup of G, denote by K G Ž N . the set of complements of N in G, i.e., K G Ž N . s L F G N LN s G, L l N s 1 4 . If M and N are normal subgroups of G, denote by K G Ž M, N . the set of subgroups of G which are complements of M and N, i.e., K G Ž M, N . s K G Ž M . l K G Ž N . . Ž2.3. PROPOSITION. Let G be a finite group and N be a minimal Ž non-tri¨ ial . abelian normal subgroup of G, isomorphic to Ž C p . n, for p prime and n ) 0. Then QG , N s X pn y < K G Ž N . < . Proof. Let H be a subgroup of G such that HN s G. Then H l N is normalized by H and by N since N is abelian. Hence either H l N s N and H s G, or H l N s 1 and H g K G Ž N .. In this case H is a maximal subgroup of G and Ž H, G . s y1. The proposition follows. Ž2.4. COROLLARY. Ž1.
Let G be a sol¨ able finite group and
1 s Ny1 - N0 - N1 - ⭈⭈⭈ - Nk s G be a chief series for G. Then for 0 F i F k, there exist a prime number pi and a positi¨ e integer n i such that NirNiy1 , Ž C p i . n i . Denote by m i the number of complements of NirNiy1 in the group GrNiy1. Then k
P˜Ž G . s
Ł Ž X pn y m i . . i
is0
i
Ž 2.5.
Ž2. Con¨ ersely, if G is a finite group, if there exists an integer k, and if there exist prime numbers pi and integers m i for 0 F i F k such that 2.5 holds, then G is sol¨ able.
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Proof. Assertion 1 follows from an obvious induction argument. For Assertion 2, suppose that G is a finite group such that Ž2.5. holds. Then the monomial of highest degree in P˜Ž G ., which is equal to P Ž G ., is the k product Ł is0 X pnii . This means that all the composition factors of G are cyclic, i.e., that G is solvable. Ž2.6. Remark. Corollary 2.4 can be viewed as a generalization of the Eulerian product formula obtained by Gaschutz ¨ w7x for the zeta function of a solvable group Žsee also w1x.. It should also be compared with the following question, cited by Brown ŽQuestion 1 of w4x.: QUESTION. If G is a finite group such that Ž G, s . has an Euler product ., is G sol¨ able? expansion with factors of the form 1rŽ1 y c i qys i 3. THE TWO NORMAL SUBGROUPS FORMULA Ž3.1. PROPOSITION w2, Lemme 20x. are normal subgroups of G then QG , N s
Ý
Let G be a finite group. If N and M
Ž L, G . P Ž L l M l N . QG r M , Ž L l N . M r M .
LFG LMsLNsG
Proof. By definition, QG , N s
Ž H , G. P Ž H l N . .
Ý HFG HNsG
As noted above
Ž H , G. s
Ž H , L . Ž L, G . .
Ý HFLFG LMsG LlHMsH
This gives QG , N s
Ý
Ý
Ž H , L . Ž L, G . P Ž H l N . .
HFG HFLFG HNsG LMsG LlHMsH
Now the conditions HN s G
HFL
LM s G
L l HM s H
are equivalent to the conditions LM s LN s G
HFL
HŽ L l N . s L
H G L l M.
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159
It follows that QG , N s
Ž L, G .
Ý LFG LMsLNsG
Ý
Ž H , L. P Ž H l N . .
HFL HGLlM Ž H LlN .sL
The inner summation is equivalent to a summation over subgroups K s HrŽ L l M . of L s LrŽ L l M . such that K. J s L, where J denotes the normal subgroup Ž L l N .Ž L l M .rŽ L l M . of L. Moreover, since HN s G, P Ž H l N . s P Ž H . P Ž N . rP Ž G . s P Ž K . P Ž N . P Ž L l M . rP Ž G . s s s
P Ž K l J . P Ž L. PŽ J .
P Ž N . P Ž L l M . rP Ž G .
P Ž K l J . P Ž L. P Ž N . P Ž J . P Ž G. PŽ K l J . PŽ L l N . P Ž L l NrL l N l M .
s PŽ K l J . PŽ L l M l N .. Thus QG , N s
Ý
Ž L, G . P Ž L l M l N .
LFG LMsLNsG
s
Ý
Ý
Ž K , L. P Ž K l J .
KFL KJsL
Ž L, G . P Ž L l M l N . Q L , J .
LFG LMsLNsG
The formula follows since L , GrM and since the image of J under this isomorphism is Ž L l N . MrM. Ž3.2. COROLLARY.
If M F N, then QG , N s QG , M QG r M , N r M .
Proof. This follows from the fact that if LM s G and M F N, then Ž L l N . M s N. This corollary also has an obvious direct proof using 1.11.
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4. I-GROUPS One of the methods used in w2x was to replace each variable X S by < S < and to look whether the resulting number QG, N is zero. This can be generalized by considering an ideal I of R and looking whether QG, N is in I or not. Ž4.1. CONVENTION. In the following, the expression ‘‘the group H is a quotient of the group G’’ means that H is isomorphic to a factor group of G. Ž4.2. DEFINITION. Let I be an ideal of R. A finite group G is called an I-group if for any non-trivial normal subgroup N of G, the polynomial QG, N belongs to I . Ž4.3. PROPOSITION. Let G be a finite group and I an ideal of R. If M and N are normal subgroups of G and GrM is an I-group, then QG , N '
Ý
Ž L, G . P Ž L l N .
Ž mod I . .
LFG LMsLNsG LlNFLlM
In particular, if QG, N f I , then GrM is a quotient of GrN. Proof. The first assertion follows from Proposition 3.1 and from the definition of an I-group. Now if QG, N f I , then there exists a subgroup L of G such that LM s LN s G and L l N F L l M. Now GrM , LrŽ L l M . is a quotient of GrN , LrŽ L l N .. Ž4.4. PROPOSITION. Let G be a finite group and I be a prime ideal of R. There exists a factor group  I Ž G . of G characterized uniquely up to isomorphism by the following properties: 1. The group  I Ž G . is an I-group. 2. If K is a quotient of G and K is an I-group, then K is a quotient of  I Ž G .. Moreo¨ er, if N is a normal subgroup of G, then the following conditions are equi¨ alent: Ža. The group  I Ž G . is a quotient of GrN. Žb.  I Ž GrN . ,  I Ž G .. Žc. QG r N f I . Proof. Properties 1 and 2 clearly show that the group  I Ž G . is unique up to isomorphism, if it exists. Let M be a normal subgroup of G, maximal subject to QG, M f I . Then by Corollary 3.2, if N is a normal subgroup of G strictly containing M, QG , N s QG , M QG r M , N r M g I .
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Since QG, M f I and since I is prime, it follows that QG r M , N r M g I . This holds for any non-trivial normal subgroup NrM of GrM, hence GrM is an I-group. By Proposition 4.3, since QG, M f I , it follows that any I-group which is a quotient of G is a quotient of GrM. Thus  I Ž G . s GrM has Properties 1 and 2. Note that by construction and by Ž1.12., if M is a normal subgroup of G such that GrM ,  I Ž G ., then QG, M f I . Now if N is a normal subgroup of G, then the group  I Ž GrN . is an I-group, which is quotient of GrN, and hence of G. Thus  I Ž GrN . is always a quotient of  I Ž G .. If Ža. holds, then  I Ž G . is an I-group, which is a quotient of GrN. Hence  I Ž G . is a quotient of  I Ž GrN ., and Žb. holds. If Žb. holds, and if MrN is a normal subgroup of GrN such that
Ž GrN . r Ž MrN . ,  I Ž GrN . , then GrM ,  I Ž GrN . ,  I Ž G .. Hence as noted above QG, M f I , and since QG, M is a multiple of QG, N , it follows that QG, N f I . Hence Žc. holds. If Žc. holds, then by Proposition 4.3 the group  I Ž G . is a quotient of GrN and Ža. holds. There are generally several normal subgroups M of G such that GrM ,  I Ž G .. They are related as follows: Ž4.5. PROPOSITION. Let G be a finite group and I be a prime ideal of R. Let M and N be normal subgroups of G such that GrM , GrN ,  I Ž G . . Let x M l N, M w G denote the poset of normal subgroups of G which contain strictly M l N and are strictly contained in M, and let n G Ž M, N . denote its reduced Euler᎐Poincare´ characteristic Ž with the con¨ ention n G Ž M, N . s 1 if M s N .. Then: 1. There exists an automorphism of the group GrM l N such that
Ž MrM l N . s NrM l N. Hence the posets x M l N, M w G and x M l N, N w G are isomorphic and in particular n G Ž M, N . s n G Ž N, M .. 2. The group MrM l N , NrM l N is isomorphic to a direct product of simple groups. 3. QG, N ' QG, M l N n G Ž M, N .< K G r M l N Ž MrM l N, NrM l N .< Žmod I ..
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Proof. Since QG, N s QG, M l N QG r M l N, N r M l N by Corollary 3.2, replacing G by GrM l N shows that it suffices to consider the case M l N s 1. In this case the groups M and N centralize each other. By Proposition 4.3, since moreover GrM , GrN implies L l M s L l N whenever LM s LN s G and L l N F L l M, QG , N '
Ž L, G .
Ý
Ž mod I . .
LFG LMsLNsG LlMsLlNs1
In other words, QG , N '
Ý
Ž L, G .
Ž mod I . .
LgK GŽ M , N .
Since QG, N f I , this shows in particular that K G Ž M, N . / ⭋. Now fix L g K G Ž M, N . and define a map from M to N by Ž m. s n if mn g L. This is well defined since L g K G Ž M, N .. Moreover, if m and m⬘ are in M, n s Ž m., and n⬘ s Ž m⬘., then mnm⬘n⬘ s mm⬘nn⬘ g L since M and N commute. This shows that Ž mm⬘. s nn⬘, hence is a group homomorphism from M to N. By symmetry, the map from N to M is defined by Ž n. s m if nm g L is also a group homomorphism, which is clearly the inverse of , since M and N commute. Moreover, the maps and are clearly L equivariant. Now define a map : G ª G by
Ž ml . s Ž m . l
᭙ m g M, ᭙l g L.
Then for m, m⬘ in M and l, l⬘ in L,
Ž mlm⬘l⬘ . s Ž m.l m⬘.l l⬘ . s Ž m . .l Ž m⬘ . .l l⬘ s Ž m . l Ž m⬘ . l⬘ s Ž ml . Ž m⬘l⬘ . . This shows that is a group homomorphism, which is clearly an automorphism of G. By construction Ž M . s N, thus induces an isomorphism of posets from x1, M w G to x1, N w G , and n G Ž M, N . s n G Ž N, M .. This shows Assertion 1. Now the maps X g w 1, N x ¬ LX g w L, G x L
Y g w L, G x ¬ Y l N g w 1, N x
L
are inverse isomorphisms of posets. Moreover, w1, N x L s w1, N x L M s w1, N xG since M and N commute. Thus Ž L, G . is equal to the reduced
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163
Euler᎐Poincare ´ characteristic of the poset x1, N w G s w1, N xG y 1, N 4, and this does not depend on the choice of L in K G Ž M, N .. Thus QG , N ' ˜ Ž x 1, N w
G
. < K G Ž M, N . <
Ž mod I .
and this proves Assertion 3. Since QG, N f I , it follows from the Crapo complementation formula that every normal subgroup of G contained in N has a complement in N, invariant by G. This can only happen if N is a direct product of simple groups, and this completes the proof of the proposition. Ž4.6. EXAMPLES. The case of b-groups discussed in w2x corresponds to the ideal I generated by all X S y < S <, for S g S . In this case if G is a p-group then  I Ž G . is trivial if G is cyclic, and is isomorphic to Ž C p . 2 otherwise. This follows from Proposition 14 of w2x, but I will give another more general proof here in Corollary 7.4. If ⌽ Ž G . is the Frattini subgroup of G, and if the order of Gr⌽Ž G . is at least p 3, then there are several normal subgroups N of G such that GrN , Ž C p . 2 , but all those subgroups contain ⌽ Ž G . and are conjugate by the automorphism group of Gr⌽Ž G ..
5. EXAMPLE: THE IDEAL OF VALUATION Ž5.1. Notation. I denote by V the ideal of R generated by all X S , for S g S. In other words, the ideal V is the ideal of polynomials with constant V is isomorphic to ⺪, term equal to zero. Clearly the quotient ring RrV thus V is a prime ideal. By definition, if N is a normal subgroup of the finite group G, then QG , N s
Ý
Ž L, G . P Ž L l N . .
L-G LNsG
In this sum, the monomial P Ž L l N . is in V , unless L l N s 1. It follows that QG , N '
Ý
Ž L, G .
Ž mod V . .
LgK GŽ N .
If N is a minimal normal subgroup of G and if N is abelian, then by Proposition 2.3 QG , N ' y< K G Ž N . <
Ž mod V .
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SERGE BOUC
In particular, QG, N g V if and only if K G Ž N . s ⭋. This is equivalent to requiring N to be contained in each maximal subgroup L of G, i.e., N to be contained in ⌽ Ž G .. Ž5.2. Notation. If G is a finite group, denote by M Ž G . the subgroup generated by all minimal normal subgroups of G. Ž5.3. PROPOSITION. Let G be a sol¨ able finite group. Then G is a V-group if and only if M Ž G . F ⌽ Ž G .. Proof. This follows clearly from the previous discussion. Ž5.4. Remark. If G is an arbitrary finite group and if M Ž G . F ⌽ Ž G ., then G is a V-group, but the converse is false in general: for example, a simple group S is a V-group if and only if Ž1, S . s 0. This happens for instance if S is the simple group of order 168. Still, M Ž S . s S is not contained in ⌽ Ž S . s 1. This remark and Proposition 4.4 suggest the following variation: Ž5.5. Notation. Denote by A the ideal of R generated by all X p for p prime. A is isomorphic to the ring ⺪wŽ X S .S g S 0 x of polynomials on Clearly RrA the set S 0 of isomorphism classes of non-abelian simple groups. Hence A is a prime ideal of R. Ž5.6. PROPOSITION. Let G be a finite group. Then G is an A-group if and only if M Ž G . F ⌽ Ž G .. Proof. If M Ž G . F ⌽ Ž G ., since ⌽ Ž G . is nilpotent all minimal normal subgroups of G are abelian and have no complement in G. If N is a minimal normal subgroup of G isomorphic to Ž C p . n, for a prime number p and a positive integer n, then by Proposition 2.3 QG , N s X pn y < K G Ž N . < s X pn g A . Hence G is an A-group. Conversely, if G is an A-group and N is a minimal normal abelian subgroup of G, isomorphic to Ž C p . n, then QG , N s X pn y < K G Ž N . < ' y< K G Ž N . < Ž mod A . Thus QG, N g A if and only if K G Ž N . s ⭋, or equivalently since N is abelian, if N F ⌽ Ž G .. Now suppose N is a non-abelian minimal normal subgroup of G. Then N , S n, for some non-abelian simple group S and some positive integer n. But QG , N s Ý Ž H , G . P Ž H l N . HFG HNsG
POLYNOMIAL IDEALS AND FINITE GROUPS
165
and the term of highest degree in this expression is obtained for H s G, and it is equal to P Ž N . s X Sn. But the ideal A consists of linear combinations of monomials Ł S g S X S␣ S for which the exponent ␣ S is positive at least for one abelian simple group S Ži.e., the monomials in which some variable X p for p prime appears.. This shows that QG, N cannot belong to A if N is non-abelian. Hence all the minimal normal subgroups of G are abelian, and it follows that M Ž G . F ⌽ Ž G ., as was to be shown. Ž5.7. COROLLARY. Let G be a finite group. Then there exists a factor group H of G, characterized uniquely up to isomorphism by the following properties: 1. M Ž H . F ⌽ Ž H .. 2. If K s GrN is a quotient of G such that M Ž K . F ⌽ Ž K ., and if M1 ᎐ G is such that GrM , H, then there exists a subgroup L of G with LM s LN s G
L l M F L l N,
and in particular K is a quotient of H. Proof. Of course H s  A Ž G .. Ž5.8. EXAMPLE. When G is a p-group, for some prime p, it is possible to describe explicitly the quotient  V Ž G . s  A Ž G .. Indeed, this quotient is obtained by considering a normal subgroup N of G, maximal subject to QG, N f V . With the notation of Proposition 2.4, if 1 s Ny1 - N0 - ⭈⭈⭈ - Nk s G is a chief series for G, and if N s Nl , for y1 F l F k, then l
QG , N s P˜Ž G . rP˜Ž GrN . s ' Ž y1 .
lq1
Ł Ž X pn y m i . i
is0
l
Ł mi
Ž mod V . .
is0
Thus N is a maximal normal subgroup such that all the m i ’s are non-zero. In particular, every minimal normal subgroup of G contained in N must have a complement in G, and hence in N. It follows that N is elementary abelian and G-semi-simple. Hence N is central. Moreover, if L irNiy1 g K G r N iy 1Ž NirNiy1 ., for 0 F i F l, then it is easy to see that L0 l L1 l ⭈⭈⭈ l L l is a complement of Nl s N in G. Thus G can be split as a direct product, G s N = L.
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Moreover, L is a V-group, thus M Ž L. F ⌽ Ž L.. Since L is a p-group, all the minimal normal subgroups of L are central of order p. It follows that no central subgroup of order p can have a complement in L. Hence N is a maximal elementary abelian central subgroup of G having a complement in G and  V Ž G . is the quotient of G by its ‘‘largest elementary abelian direct summand.’’ In this case, one can say exactly how many subgroups N such that GrN ,  V Ž G . there are: indeed, with the previous notations, the group M Ž L. is equal to the subgroup ⍀ 1Ž ZŽ L.. generated by the central elements of order p of L. But ⌽ Ž G. s 1 = ⌽ Ž L. and ⍀ 1Ž Z Ž G . . s N = ⍀ 1Ž Z Ž L . . .
Ž 5.9.
Taking the intersection of those two equations gives ⌽ Ž G . l ⍀ 1Ž Z Ž G . . s 1 = ⍀ 1Ž Z Ž L . . . It follows from 5.9 that N must be a complement of ⌽ Ž G . l ⍀ 1Ž ZŽ G .. in ⍀ 1Ž ZŽ G ... Conversely, if N is such a complement, there exists a subgroup K of G, containing ⌽ Ž G ., such that Kr⌽Ž G . is a complement of N⌽ Ž G .r⌽ Ž G . in Gr⌽Ž G ., since Gr⌽Ž G . is elementary abelian. In other words, KN s G
K l N⌽ Ž G . s ⌽ Ž G . .
It follows that K l N F N l ⌽ Ž G . s 1, thus K is a complement of N. Clearly now N is G-semi-simple, and it follows that QG, N f V . Thus the subgroups N such that GrN ,  I Ž G . are exactly the complements of ⌽ Ž G . l ⍀ 1Ž ZŽ G .. in ⍀ 1Ž ZŽ G ... 6. DIRECT PRODUCTS Ž6.1. Notation. Let G and H be finite groups. Denote by p1 and p 2 the projections from G = H to G and H respectively. If L is a subgroup of G = H, set p1 Ž L . s g g G N ᭚h g H , Ž g , h . g L4 k 1 Ž L . s g g G N Ž g , 1 . g L4 p 2 Ž L . s h g H N ᭚ g g G, Ž g , h . g L4 k 2 Ž L . s h g H N Ž 1, h . g L4 .
POLYNOMIAL IDEALS AND FINITE GROUPS
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Then k i Ž L. 1 ᎐ pi Ž L. for i s 1, 2, and the groups q Ž L. s LrŽ k 1Ž L. k 2 Ž L.., Ž . Ž . p1 L rk 1 L , and p 2 Ž L.rk 2 Ž L. are canonically isomorphic. Let G and H be finite groups. Then
Ž6.2. PROPOSITION.
P˜Ž G . P˜Ž H . s
P˜Ž L . .
Ý LFG=H p 1Ž L .sG p 2 Ž L .sH
Proof. This is a consequence of the definition of the polynomials P˜ by Mobius inversion. Indeed, for any finite groups G and H ¨ PŽG = H . s
P˜Ž L . .
Ý
Ž 6.3.
LFG=H
Now setting
Ž G, H . s
Ý
P˜Ž L . ,
LFG=H p 1Ž L .sG p 2 Ž L .sH
the right-hand side of Eq. Ž6.3. can be written as
Ž A, B . .
Ý AFG BFH
Thus
Ý Ž C, G . Ž D, H . P Ž C = D . CFG DFH
Ý Ý Ž C, G . Ž D, H . Ž A, B .
s
CFG AFC DFH BFD
s
Ý AFG BFH
ž
Ý
Ž C, G . Ž D, H . Ž A, B .
AFCFG BFDFH
/
s Ž G, H . . The proposition follows, since the left-hand side is equal to P˜Ž G . P˜Ž H ., because P Ž C = D . s P Ž C . P Ž D . for any C F G and D F H. Ž6.4. COROLLARY. group, then
If G and H ha¨ e no non-tri¨ ial isomorphic factor P˜Ž G = H . s P˜Ž G . P˜Ž H . .
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Proof. Indeed, in this case the only subgroup L of G = H such that p1Ž L. s G and p 2 Ž L. s H is G = H itself, since q Ž L. is a quotient of both G and H, hence it is trivial. Ž6.5. PROPOSITION w2, Lemme 22x.
P˜Ž G = H . s P˜Ž G . P˜Ž H .
Let G and H be finite groups. Then
Ý LFG=H p 1Ž L .sG p 2 Ž L .sH
˜ Ž x 1, q Ž L . w
qŽ L.
P˜Ž q Ž L . .
..
Proof. Denote by K the group G = H and by M and N the normal subgroups G = 1 and 1 = H of K. Then by Proposition 3.1 since M l N s1 QK , N s
Ž L, K . Q K r M , Ž L l N . M r M .
Ý LFK LMsLNsK
The condition LM s LN s K is equivalent to p1Ž L. s G and p 2 Ž L. s H. In this case, moreover, the maps X g x L, K w ¬ k 1 Ž X . rk 1 Ž L . g x 1, Grk 1 Ž L . w Yrk 1 Ž L . g x 1, Grk 1 Ž L . w
G
G
¬ Ž g , h . g G = H N ᭚a g G, Ž a, h . g L, gay1 g Y 4 are mutual inverse isomorphisms of posets. Since Grk 1Ž L. , q Ž L., it follows that Ž L, G . s ˜ Žx1, q Ž L.w qŽ L. .. Moreover, Q K r M , Ž L l N . M r M s Q L r L l M , Ž L l N .Ž L l M .r L l M s
P˜Ž LrL l M . P˜Ž Lr Ž L l N . Ž L l M . .
.
But LrL l M , KrM , H and Lr Ž L l N . Ž L l M . s Lr Ž k 1 Ž L . = k 2 Ž L . . s q Ž L . .
POLYNOMIAL IDEALS AND FINITE GROUPS
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It follows that
Q K , N s P˜Ž H .
Ý LFG=H p 1Ž L .sG p 2 Ž L .sH
˜ Ž x 1, q Ž L . w
qŽ L.
P˜Ž q Ž L . .
..
The proposition follows, since Q K , N s P˜Ž K .rP˜Ž KrN . s P˜Ž K .rP˜Ž G .. Ž6.7. COROLLARY. Let G and H be finite groups, and denote by Gs , Hs , and Ž G = H .s , Gs = Hs the respecti¨ e largest semi-simple quotients of G, H, and G = H. Then P˜Ž G = H . P˜Ž Ž G = H . s .
s
P˜Ž G . P˜Ž H . P˜Ž Gs . P˜Ž Hs .
.
Proof. The only non-zero terms in the formula Ž6.6. correspond to subgroups L of G = H such that q Ž L. is semi-simple. Moreover, since p1Ž L. s G and p 2 Ž L. s H, the group q Ž L. is a quotient of G and H. It follows that if M and N are normal subgroups of G and H respectively, such that GrM , Gs and HrN , Hs , then L G M = N. Then the group L⬘ s LrM = N is a subgroup of Gs = Hs , Ž G = H .s and q Ž L . s Lr Ž k 1 Ž L . = k 2 Ž L . . , Ž LrM = N . r Ž k 1 Ž L . = k 2 Ž L . rM = N . , L⬘r Ž Ž k 1 Ž L . rM . = Ž k 2 Ž L . rN . . s q Ž L⬘ . . The corollary follows, since the correspondence L ¬ L⬘ from
K F G = H N p1Ž K . s G, p2 Ž K . s H , K G M = N 4 to the set
K ⬘ F Gs = Hs N p1Ž K ⬘. s Gs , p2 Ž K ⬘. s Hs 4 is clearly one to one. Corollary 6.7 gives a way to compute P˜Ž G = H . knowing P˜Ž G ., P˜Ž H ., and the groups Gs and Hs . The following notation will be convenient: Ž6.8. Notation. If G and H are finite groups, denote by sŽ G, H . the number of surjective group homomorphisms from G to H.
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If n, m g N, set ny1
F Ž G, n . s
Ł Ž P˜Ž G . y s Ž G i , G . .
is0
B Ž G, m, n . s
F Ž G, m q n . F Ž G, m . F Ž G, n .
.
Here the letter F stands for ‘‘factorial’’ and the letter B for ‘‘binomial.’’ Note that B Ž G, m, n. is an element of the field of fractions of R. Ž6.9. PROPOSITION. Ž1. If S is a finite simple group and n g N, then P˜Ž S n . s F Ž S, n.. Moreo¨ er, sŽ S n, S . is equal to p n y 1 if S , C p , and to n
Ł B Ž S, aS , bS . .
Sg S
Proof. The first formula follows from an easy induction argument, using Proposition 6.2 or Ž6.6.. Using this, the second formula follows from Corollaries 6.4 and 6.7. Ž6.10. LEMMA. Let I be a prime ideal of R. If G and H are I-groups, and if they ha¨ e no non-tri¨ ial isomorphic factor group, then G = H is an I-group. Proof. Since G and H are quotients of G = H, it follows that they are quotients of  I Ž G = H .. Hence there is a group homomorphism
: I Ž G = H . ª G = H such that p1 ( and p 2 ( are surjective. Let K denote the image of . Then p1Ž K . s G and p 2 Ž K . s H. Thus q Ž K . is a quotient of both G and H, hence it is trivial. It follows that K s G = H, hence  I Ž G = H . , G = H. Thus G = H is an I-group. Ž6.11. PROPOSITION. Let I be a prime ideal of R, and G and H be finite groups ha¨ ing no non-tri¨ ial isomorphic factor group. 1.
If M 1 ᎐ G and N 1 ᎐ H, then QG= H , M=N s QG , M Q H , N .
2.
Moreo¨ er,
I Ž G = H . , I Ž G. = I Ž H . .
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3. The group G = H is an I-group if and only if G and H are I-groups. Proof. Let M be a normal subgroup of G and N be a normal subgroup of H. Then by Corollary 6.4, since GrM and HrN have no non-trivial isomorphic factor group, P˜Ž G = H . s P˜Ž G . P˜Ž H . P˜Ž G = HrM = N . s P˜Ž GrM = HrN . s P˜Ž GrM . P˜Ž HrN . . Taking the quotient of those equations gives Assertion 1, QG= H , M=N s QG , M Q H , N . Clearly  I Ž G . and  I Ž H . have no common non-trivial factor group. Hence by Lemma 6.10  I Ž G . =  I Ž H . is an I-group, and it is a quotient of G = H. Hence it is a quotient of  I Ž G = H .. Conversely, if M 1 ᎐ G is such that GrM ,  I Ž G . and N 1 ᎐ H is such that HrN ,  I Ž H ., then QG, M f I and Q H , N f I , thus QG=H , M=N f I by Assertion 1 since I is prime. By Proposition 4.3 the group  I Ž G = H . is a quotient of
Ž G = H . rŽ M = N . ,  I Ž G. =  I Ž H . . Assertion 2 follows. Finally, G = H is an I-group if and only if G = H s  I Ž G = H .. By Assertion 2, this is equivalent to G s  I Ž G . and H s  I Ž H ., which proves Assertion 3.
7. NILPOTENT I-GROUPS It is possible to describe all the nilpotent I-groups when I is a prime ideal of R. Since any nilpotent group is isomorphic to the direct product of its Sylow p-subgroups, for prime numbers p, it follows from Proposition 6.11 that a nilpotent group is an I-group if and only if all its Sylow subgroups are I-groups. Thus in order to describe the nilpotent I-groups, it suffices to describe the p-groups which are also I-groups. Ž7.1. PROPOSITION. Let I be a prime ideal of R, let p be a prime number, and let G be a finite p-group. Then 1. If G is elementary abelian of order p n, then G is an I-group if and only if n s 0 or n G 1 and X p y p ny 1 g I .
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2. If G is not elementary abelian, then G is an I-group if and only if X p g I and one of the following holds: Ža. p g I Žb. ⍀ 1Ž ZŽ G .. F ⌽ Ž G ., or equi¨ alently, the group G cannot be written G , C p = H, for some finite group H. Proof. Suppose G is non-trivial and let N be a minimal normal subgroup of G. Then N is central of order p and QG , N s X p y < K G Ž N . < , since any proper subgroup L of G such that LN s G is a complement of G, and L is a maximal subgroup of G in this case. Suppose that G is elementary abelian of order p n. If n s 0, then G is trivial and G is an I-group. If n G 1 and if N is a subgroup of G of order p, then QG , N s X p y p ny 1 since there are p ny 1 complements of N in G. This proves Assertion 1. If G is not elementary abelian, then there exists a central subgroup N of G of order p contained in ⌽ Ž G .. If N is such a subgroup, then K G Ž N . s ⭋ and QG , N s X p .
Ž 7.2.
Now if M is a central subgroup of G of order p and M g ⌽ Ž G ., then QG , N s X p y < K G Ž M . < .
Ž 7.3.
Moreover, in this case K G Ž M . / ⭋, thus there exists a group H such that G , M = H and < K G Ž M . < s
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Proposition 7.1 can be reformulated as follows: Ž7.4. COROLLARY. Let I be a prime ideal of R, let p be a prime number, and let G be a finite p-group. Then: If X p g I and p g I , then G is an I-group if and only if G ` C p . If X p g I and p f I , then G is an I-group if and only if ⍀ 1Ž ZŽ G .. F ⌽ Ž G .. If X p f I , then let h, k g ⺞ j ⬁4 defined by 䢇 䢇
䢇
h s Inf l g ⺞ N X p y p l g I 4 k s Inf l g ⺞ y 0 4 N 1 y p l g I 4 . Then G is an I-group if and only if G is elementary abelian of order p n with n s 0, n ' h q 1 Ž k . if h / ⬁ and k / ⬁, or n s h q 1 if h / ⬁ and k s ⬁. Proof. Indeed, if X p g I and p g I , then by Proposition 7.1 the only case where G is not an I-group is the case G , C p . If X p g I and if a non-trivial elementary abelian p-group of order p n is an I-group, then p ny 1 g I , hence 1 g I or p g I . Thus if p f I , then no non-trivial elementary abelian p-group can be an I-group, and G is an I-group if and only if ⍀ 1Ž ZŽ G .. F ⌽ Ž G .. Finally, if X p f I , then G is an I-group if and only if G is elementary abelian of order p n with n s 0 or n G 1 and X p y p ny 1 g I . This cannot happen if h s ⬁, and if h g ⺞ this is equivalent to p h y p ny 1 s p h Ž 1 y p ny hy1 . g I . But p h f I , since otherwise X p s Ž X p y p h . q p h g I , and this is equivalent to 1 y p ny hy1 g I , or n ' h q 1Ž k ., which has to be viewed as an equality if k s ⬁. Ž7.5. Remark. The last case of Corollary 7.4 should be compared with Theorem 8.2 of w3x, which deals with ‘‘b-groups in characteristic q,’’ i.e., with the case where I is generated by a prime number q / p Žor q s 0., and by X S y < S <, for all simple groups S. REFERENCES 1. S. Bouc, Homologie de certains ensembles ordonnes, ´ C. R. Acad. Sci. Paris Ser. I 299, No. 2 Ž1984., 9᎐12. 2. S. Bouc, Foncteurs d’ensembles munis d’une double action, J. Algebra 183 Ž1996., 664᎐736; doi: 10116.jabr.1996.0236.
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3. S. Bouc and J. Thevenaz, The group of endo-permutation modules, In¨ ent. Math. 139 ´ Ž2000., 275᎐349. 4. K. Brown, The coset poset and the probabilistic zeta function of a finite group, preprint, 1999. 5. K. Brown and J. Thevenaz, A generalization of Sylow’s third theorem, J. Algebra 115 ´ Ž1988., 414᎐430. 6. H. Crapo, The Mobius function of a lattice, J. Combin. Theory 1 Ž1966., 126᎐131. ¨ 7. W. Gaschutz, Gruppen, Illinois J. Math. 3 ¨ Die Eulersche Funktion endlicher auflosbarer ¨ Ž1959., 469᎐476. 8. P. Hall, The Eulerian functions of a group, Quart. J. Math. 7 Ž1936., 134᎐151.