Chapter 20 Induction Theorems and Applications

Chapter 20

Induction Theorems and Applications A principal objective of this chapter is to establish a number of classical results pertaining to induced characters. These include Brauer’s characterization of characters, a converse of Brauer’s theorem, necessary and sufficient conditions for a class function to be an R-generalized character ( R is a subring of (c ), rational valued characters, etc. The main tool in our investigation will be the Witt-Berman’s induction theorem and its converse discovered by Berman. As one of the many applications, we establish an elegant result due to Turull (1982) which asserts that if V and W are Q G-modules such that (Cv(g))= d i m q (Cw(g))for all g E G , then V 2 W .

dim^

1

The Witt-Berman’s induction theorem

The discussion below will require a considerable amount of notation. Therefore it will be especially useful t o assemble most of it in one place. G is a finite group of exponent n. F is an arbitrary field of characteristic 0. E~ is a primitive m-th root of 1 over F. R is the integral closure of Z in F ( E ~ ) . Im is the multiplicative group consisting of those integers p , taken mod) F. ulo m ,for which E~ H ELdefines an automorphism of F ( E ~over C ~ R ( G , Fis) the ring of all R-linear combinations of F-characters of G (here each F-character of G is regarded as a class function from G to F ( E ~ ) ) . Ch(G,F ) is the ring of all generalized F-characters of G. 779

Induction Theorems and Applications

780

The following lemma due to Solomon (1961a, Lemma 2) will play an important role in the proof of the main result.

Lemma 1.1. Let H =< g > K be a semidirect product of the normal cyclic subgroup < g > of order m and a group K such that for each x E K there exists p E 1, such that xgx-' = g p , Denote by 5' the integral closure of Z in Q ( E ~ ) . Then there exists an S-linear combination X of F-characters o f H such that m if i E Im X(gi) = 0 if i #Im Proof. Let p :< g >+ Z be given by p(gi) = m if i E 1, and p(gi) = 0 if i # I,. Let xo, .. .,xrn-l be all irreducible Q (&,)-characters of < g >. Since p is a class function on < g >, it follows from Theorem 19.3.5 that

c

m-1

p=

where

cixi

i=O

m-1

Then ci E S and each 0 E Gal(F(E,)/F) fixes ci. Since u fixes p and permutes the x;,we see that the F-conjugate characters in (1)appear with the same coefficients. Thus, by Theorem 19.4.2 (i), p is an S-linear combination of irreducible F-characters of < g >, say p = s1a1 t * - .4- s k a k , 3; E s, ai is an irreducible F-character of G. By Corollary 19.2.10, each ai can be extended to an F-character pi of H. Hence X = s& t - .- t $& satisfies the desired property. Let H be a subgroup of G and let p be a prime. We say that H is Felementary with respect to p if the following two conditions hold : (i) H is a semidirect product < g > Ii' of the normal cyclic subgroup < g > of order m coprime to p and a pgroup K . (ii) For any x E K ,there exists p E I , such that zgx-' = g p . We say that G is F-elementary if G is F-elementary with respect to some prime p. To describe a typical situation in which F-elementary subgroups arise, we need the following definition. Let g E G. Then the F-normalizer N of g is defined to be

N = ( 5 E G1zgx-l = g p for some p E Im}

1 The Witt-Berman’s induction theorem

78 1

where m is the order of g .

Lemma 1.2. Let N be the F-normalizer of g E G , let p be a prime and let K be a Sylow p-subgroup of N . Then (i) N is a subgroup of G and < g > d N . (ii) If g is p-regular, then H =< g > K is an F-elementary (with respect to p) subgroup of G. Proof. This is a direct consequence of the definitions. W

Lemma 1.3. Let g E G be an element of order m and let 5’ be an integrd domain containing Z [ E ~ ] .If P is a prime ideal of S containing a prime p , then for any F-character x of G, x ( g ) f X(gpl)(modP ) . where ~k and E; are the p and p’-parts of Proof. Write E~ = ELEL, E ~ Then . E~ = ER(modP) since ~k = l ( m o d P ) . On the other hand, since k gpt = g9 it follows from Corollary 17.1.11 that we may write x ( g ) = EZ and k

for some t i

2 1. Hence x ( g ) = X(gpt)(modP ) as required.

The following result is known as the Witt-Berman’s Induction Theorem.

Theorem 1.4. (Witt (1952), Berman (1956b)). Let G be a finite group and let F be an arbitrary field of characteristic 0. Then any F-character of G can be written in the form where each z; E group of G.

Z and each x; is a n F-character of an F-elementary sub-

Proof. (Solomon (1961a). For the sake of clarity, we divide the proof into a number of steps. Step 1. Given a subgroup H of G , let [ C ~ R ( H , F )C] C ~ ~R(G,F) consist of all R-linear combinations of F-characters of G induced from F characters of H . Then, by Proposition 19.1.7 (i)? [ChR(H,F)]’ is an ideal

Induction Theorems and Applications

782

of C ~ R ( GF ), . NOW put

where H runs over all F-elementary subgroups of G with respect to distinct F ), . primes dividing the order of G. Then VR(G,F ) is an ideal of C ~ R ( G Similarly, if VZ ( G , F ) is defined as VR(G,F)by replacing R by Z , then V z (G,F) is an ideal of the character ring Ch(G,F). We wish to show that V z (G,F) = Ch(G,F ) which will imply the assertion of the theorem. Denote by 1~ the trivial character of G, i.e. 1G(g) = 1 for all g E G. In this step we claim that it sufices to show that 1~E VR(G,F). Assume that lc E VR(G,P).Then 1~ = El=,T ~ Q ; ,T; E R, a;is induced from an F-character of an elementary subgroup of G . Put r o = 1 and R. Since the additive group consider the Z5 -module & = Cf=, 22 ~i of Ro is torsion-free and finitely generated, Ro is Z -free of finite rank. Moreover, Ro/Z is a torsion-free (hence free) Z -module. Thus we may = TO = 1. Hence choose a Z-basis {PI,.. . , P k } of l& with k

1~= CPjpj j=1

(pj E

vz

(

~

9

~

1

)

where = 1and P I , . ..,p k are Q -linearly independent elements of R. Now each P j is a Z -linear combination of all irreducible F-characters y1,. . . ,ys of G. Writing p j = Zjmy,, zjm E Z , 1 5 j 5 k, we then have

Because 71,. . . ,ys are F-linearly independent, we obtain

Bearing in mind that

PI,, . . ,P k

m=l

are Q -linearly independent, it follows that

1 The Witt-Berman's induction theorem

783

Since V z ( G ,F ) is an ideal of C h ( G , F ) ,it follows that VZ (G, F) = Ch(G,F), as claimed. Step 2. Let p be a prime and let g E G be pregular. Our aim here is to exhibit $ = & E VR(G,F)for which the following two properties hold : (i) $(z) = 0 if z E G is pregular and 2 is not F-conjugate to g. (ii) $(g) E Z and $(g) = l(modp). To begin with, we denote by N the F-normalizer of g in G. By Lemma 1.2, N is a subgroup of G with < g > d N . Furthermore, if K is a Sylow p-subgroup of N , then H =< g > K is an F-elementary (with respect t o p) subgroup of G with IHI = mt, where m is the order of g and t is the order ) XG E VR(G,F). of I<. Let X be as in Lemma 1.1; hence X E C ~ R ( G , Fand Extend X to G by setting X to be 0 on G - H . Then

P(z) = m-'t-'

c

X(y-'zy)

for all z E G

Y€G

Now each pregular element z E H belongs to < g >. Hence, for any pregular element z E G with y-'zy #< g >, we have X(y-'zy) = 0. On the then by definition of A, X(y-'zy) = 0. other hand, if y-'zy = g p with p # Im, Since the equality y-lzy = g p for some p E Im implies that z and g are F-conjugate, we deduce that XG is 0 on all p-regular elements of G which are not F-conjugate to g. Finally, since X(y-'gy) # 0 implies y-lgy = g p for some p E I,, i.e. y E N ,we have

Hence by choosing z E 22: with z ( N : K )= l(modp),the function II,= zXG satisfies the required properties. Step 3. Let p be a fixed prime dividing n and let P be a prime ideal of R containing p. We wish to find Q E VR(G,F)such that for all g E G, a(g) l ( m o d P ) . To this end, we first note that, by Lemma 1.3, x ( g ) = x(g,t)(modP) for all g E G and all F-characters x of G. Hence the same congruence holds for all a E C ~ R ( G F ), . Let {gl, . . .,gk} be all representatives for the F-conjugacy classes of pregular elements of G. Then, by Step 2, for each i E (1,. . . ,I c } , there exists $i E VR(G,F)such that

=

+,i(gi)

= l ( m o d P ) ,+;(gj)

= 0 for j

#i

Induction Theorems and Applications

784

Set c t = $ ~ i - . - - + $ % . Thencr(gi)= l ( m o d P ) f o r a l l i ~(1, ...,k}. Finally, given g E G, g,! is F-conjugate to some gi. Hence, by Lemma 17.5.5 a(g)

= cY(gpt)I a(gi) = 1(modP)

as required. Step 4. Completion of the proof. By Step 1, it suffices to show that 1~E VR(G,F).Fix a prime p dividing IGl and write IGl = pks with ( p , s ) = 1. Then it suffices to show that S ~ E G VR(G,F ) . Let cy E VR(G,F ) be as in Step 3. Then

Our construction of cy did not depend on the choice of the prime ideal P containing p and so choosing a sufficiently large t , we have +)pt

= l(modpk~)

for all g E G

Now put 8 = S ( ~ G- a p t ) E C ~ R ( G , F )We . claim that 0 E VR(G,F);if sustained, it will follow that S ~ EG VR(G, F ) (since sap' E VR(G, F)),which will complete the proof. Let 91,. . . ,g, be the representatives for the F-conjugacy classes of G. For each j E (1,. ..,r } , let pj be defined on < gj > as follows : pi($) = mj if i E Im,and pj(g$) = 0 otherwise, where mj is the order of gj. As we have seen in the proof of Lemma 1.1, each pj is an R-linear combination of irreducible F-characters of < gj > and so each py E VR(G,F ) . Furthermore, we know that each py is 0 on all elements of G which are not F-conjugate to gj. An easy calculation shows that

where Nj is the F-normalizer of gj E G. Moreover,

Indeed, since 0 and p?, 1 5 j 5 T , are constant on F-conjugacy classes of G, it suffices to verify (3) for each gi, and this is true by virtue of (2). Finally, we know that e(gj) = s ( 1 - cy(gj)p') E O(modlG1R)

1 The Witt-Berman’s induction theorem

785

Hence all the coefficients in the right-hand side of (3) belong t o R. Thus 8 E V R ( G , F )and the result follows. I

To prove an application of Theorem 1.4, we need the following lemma. Lemma 1.5. Let H be a subgroup of G and let S be the ring of all functions f : G F satisfying the following properties : (i) f is constant on F-conjugacy classes of G. (ii) fH E C h ( H , F ) Then S 2 C h ( G , F ) and, for any ideal I of C h ( H , F ) , I n d s ( 1 ) = {aG\aE I } is an ideal of S . --$

Proof. That S 2 C h ( G , F ) is a consequence of Lemma 17.5.5. Given I , we have aG - PG = (a- P)G E Ind$(I). Let f E S and a E I . Then f E C f ( G ,F ) , a E C f ( H ,F ) and fHa E I . Hence, by Lemma 19.3.2 (i), ( f ~ a= )f a~G E Indg(1) as required. I a,P E

Theorem 1.6. (Berman (1958)). Let f : G F be any map. Then f E C h ( G ,F ) if and only if the following conditions hold : (i) f is constant on F-conjugacy classes o j G . (ii) For any F-elementary subgroup H of G, fH E C h ( H ,F ) . --f

Proof. Let S be the ring of all functions f : G + F satisfying (i) and (ii). Denote by I the 23 -submodule of C h ( G ,F ) generated by F-characters which are induced from F-characters of F-elementary subgroups of G. By Lemma 1.5, S _> Ch(G,F ) and I is an ideal of S. Since, by Theorem 1.4, 1 = C h ( G ,F ) we have S = Ch(G,F ) . W We close by proving a converse of Theorem 1.4.

..,H , be S U ~ ~ ~ OofU Theorem 1.7, (Berman (1958)). Let H I ,Hz,, G such that each F-character of G is an integral linear combination of characters induced from F-characters of the subgroups Hi. Then, for any F elementary subgroup H of G, there exists Hi, i E (1,. . . ,n } which contains a conjugate of H .

Proof. By hypothesis, H is a semidirect product < g > K of the normal cyclic p’-subgroup < g > and a p-group K (where p is a prime).

~ S

Induction Theorems and Applications

786

Furthermore, K is contained in the F-normalizer N of g in G. Choose a Sylow psubgroup P of N containing K . Then < g > P is an F-elementary subgroup of G containing H . It will be shown that there exists Hi which contains a conjugate of E =< g > P, which will complete the proof. Let x i j be an irreducible F-character of H ; . Then, by Proposition 19.1.2,

where 61,. . .,b, are the representatives for the conjugacy classes of Hi contained in the conjugacy class of G containing g and z i j t = ( C ~ ( b t: )C ~ , ( b t ) ) (by convention, each x i j ( b t ) = 0 if there are no such representatives). Let Nit be the F-normalizer of bt in Hi and let Nt be the F-normalizer of bt in G . We claim that zijt = ( N t : N i t ) (5) Indeed, the set of all elements z in G with z-'btz = b r , for a fixed p , forms a coset of G with respect to CG(bt). Therefore lNtl = l C ~ ( b t )where l~ s is the number of elements in < bt > which are F-conjugate to bt. Similarly, IN i t I = ICH,(bt )IS proving (5). By hypothesis, we may write 1 = Ci,ja i j X i j with a;j E Z and so, by (41, zijt f O(modp) for some i , j , t (6) It follows from (5) and (6) that some Nit contains a Sylow p-subgroup Q of Nit C H;.Since bt = z-lgz for some 2 E G, we Nt. Hence < bt > Q deduce that z-'Pz is a Sylow p-subgroup of N t . Thus y - ' ( ~ - ~ P z ) y = Q for some y E Nt. Therefore (sy)-'E(zy) =< bt > Q C Hi and the result follows. m

2

Brauer's theorems

In this section, we derive a number of fundamental results of Brauer as a consequence of the Witt-Berman's induction theorem. Let G be a finite group and let p be a prime. We say that G is pelementary if G is a direct product of a cyclic group and a pgroup. We also say that G is elementary if it is pelementary for some prime p . Observe that elementary groups are nilpotent and hence (by Corollary 18.12.4) M groups. By a linear character , we understand a character of degree 1.

2 Brauer’s theorems

787

Theorem 2.1. (Bmuer’s Characterization of Characters, Brauer (1947), (1953)). (i) Any C -character of G is an integml linear combination of characters induced from linear characters of suitable elementary subgroups of G. (ii) Let a : G 4 C be a class function. Then CY is a generalized character of G if and only if for every elementary subgroup E of G, CYE is a generalized character of E . (iii) Let a : G -, C be a class function. Then CY is an irreducible (I= chamcter i f and only if the following properties hold : (u) For every elementary subgroup E of G, CYEis a generalized character of E . (b) 1GI-l C g E G 4 M g - l ) = 1 (c) a(1)> 0.

Proof. (i) It is clear that a subgroup E of G is elementary if and only if it is (I: -elementary. Now apply Theorem 1.4 and the fact that, by Corollary 18.12.4, elementary groups are M-groups. (ii) This is a special case of Theorem 1.6 in which F = (I: . (iii)Apply (ii) and Lemma 19.3.7. (iii). H Theorem 2.2. ( A converse to Brauer’s Theorem, Green (1955)). Let H I , . ..,H , be subgroups of G such that any (I: -chamcter of G is an integral linear combination of (c -characters induced from the characters of the subgroups Hi,1 5 i 5 n. Then, for any elementary subgroup E of G, there exists i E ( 1 , . . . ,n ) such that Hi contains a conjugate of E .

Proof. This is a special case of Theorem 1.7 in which F = (I:. Theorem 2.3. (Bmuer (1945), (1947)). Let G be a finite group of exponent n and let F be a field over which the polynomial X n- 1 splits into linear factors. Then F is a splitting field for G.

Proof. The case charF = p > 0 is a consequence of Proposition 17.2.5. Assume that charF = 0. Then, by hypothesis, F 2 Q ( E ) where E is a primitive n-th root of 1. Hence, by Proposition 17.2.2, it suffices to show that Q ( E ) is a splitting field for G. Let V be a simple (I: G-module which affords the (I:- character x and let F = Q ( E ) . By Proposition 11.1.11, it suffices to show that V is realizable

Induction Theorems and Applications

788

over F . Since x ( g ) E F for all g E G, we have F ( x ) = F . Hence it suffices is equal to 1. By Theorem 2.1 (i), we to show that the Schur index rn&) may write

x = Cz;xG

zi E

z

(1)

where {xi} are linear characters of elementary subgroups of G. It is clear that each X? is afforded by some FG-module. By Theorem 14.4.1 (i), the multiplicity of x in each character xf is a multiple of r n ~ ( x )On . the other = 1 and the hand, by (l),the multiplicity of x in Czixf is 1. Thus rn&) result follows. H For future use, we need an easy generalization of Theorem 2.1 (ii). Let R be a subring of (c , By an R-generalized character of G, we understand an R-linear combination of irreducible (c -characters of G.

Theorem 2.4. Let LY : G --t (c be a class function and let R be a subring of (c . Then Q is an R-generalized character of G if and only i f for any elementary subgroup E of G, Q E is an R-genemlized character of E .

Proof. Let S be the subring of Cf (G) consisting of all p such that PE is an R-generalized character of E for any elementary subgroup E of G. Denote by K the subring of Cf(G) consisting of all R-generalized characters of G. Then obviously S 2 K . Let I be the R-submodule of K generated by all (I:-characters which are induced from (c -characters of elementary subgroups of G. If X E S and x is a (c -character of E , then X - xG = ( X E X ) ~(Lemma 19.3.2 (i)). Hence I is an ideal of S. Since, by Theorem 2.1 (i), 1~ E I we have S = K , as desired. H To prove our find result, we need the following application of Theorem

2.4.

Lemma 2.5. Let IGl = p"k with ( p , k ) = 1, p-prime, let g be a p regular element of G and let R be the ring of algebraic integers in (Q ( E ) , where E is a primitive k-th root of unity. Define Q : G (I: by --f

if xp1 is conjugate to 9 otherwise Then a is an R-generalized character of G.

3 Rational valued characters

789

Proof. Let E be an elementary subgroup of G. By Theorem 2.4, it suffices to show that C X Eis an R-generalized character of E . Write E = El x E2 where El is a pgroup and E2 is a $-group. If g is not conjugate t o an element of E2, then a~ = 0 which implies the result. Assume that gl, . . . , g r are all distinct elements of E2 which are conjugate to g . Then the elements x E E with xpt conjugate to g are precisely the elements of the cosets E l g i , 1 5 i 5 r . Let x be an irreducible (I: -character of E . Then, by Proposition 17.8.1, x = x1 x x 2 where x i is an irreducible (c -character of Ei, i = 1,2. Hence

cc T

IEI < Q E , X >=

T

4 t g ; ) ; y ( t g i= )

i=l tcE1

If

XI

#l

~then ~

kCXz0 i=l

c

Xl(4

tEEi

CtEEl , x l ( t ) = 0 and so < C Y E , X>= 0. If x i

= l ~then~ ,

T

i=l

Since 1EI = desired.

l E l l JE2l and

lE2l divides k, it follows that < CYE,X> E R , as

Corollary 2.6. Let IGI = pnk with ( p , k ) = 1, p prime, let R be the ring of algebraic integers in Q ( E ) , where E is a primitive (GI-th mot of unity and let P be a prime ideal of R with p E P . Then, f o r any given x,y E G, xp: and yp: are conjugate if and only if for any irreducible (c -character x of G, x ( 4 = X ( Y > (mod PI Proof. Assume x(x) G X(y)(rnodP ) for all irreducible (I: -characters x of G. Let a : G + (I: be defined by a(g)= k if gp: is conjugate to xp: and otherwise a ( g ) = 0. Then, by Lemma 2.5, is an R-generalized character of G and so a ( . ) 3 a ( y ) ( m o d P ) .By definition, a(.) = k. If a ( y )= 0, then k E P which is impossible since p E P and ( p , k ) = 1. Thus a(y) = k and hence ypt is conjugate to z p t . The converse being a consequence of Lemma 1.3, the result follows.

3

Rational valued characters

In this section, we prove a classical result of Artin concerning rational valued complex characters of finite groups and provide some applications. In what

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790

follows, G denotes a finite group and all characters are assumed to be C characters. We say that a character x of G is rational valued if

x(g) E Q

for all 9

EG

It is clear that any Q-character is rational valued, but the converse need not be true. Given a subgroup H of G, we write 1~ for the principal character ) 1 for all h E H ) . For any cyclic subgroup H of G , of H (i.e. 1 ~ ( h = a ( H ) : H + Z is defined by a ( H ) ( z )= JHJ if H =< 2 > and cr(H)(z)= 0 otherwise. It is clear that a ( H )is a class function.

Lemma 3.1. Let H be a cyclic subgroup of G. Then (i) a(H) is an integral linear combination of characters of H induced from principal chamcters of cyclic subgroups of H . (ii) For any given g E G, c r ( H ) G ( g ) = ~ N G ( H ) Iif g is conjugate to a generator of H and a ( H ) G ( g ) = 0 , otherwise.

Proof. (i) The case H = 1 being obvious, we argue by induction on / H I . To this end, we first show that a(s)H =

*

1H

SCH

where S ranges all subgroups of

H.For each z E H , we have

where, by convention, a ( S ) ( y ) = 0 for y E H - S. Since a ( S ) ( z )= 0 if S #< 2 > and a ( S ) ( z ) = IS) if S =< z >, we have ‘ & C-H a ( S ) H ( ~ ) = \HI, proving (1). It follows from (1) that

a(H) = I H l l H -

C CK(S)~

SCH

Applying the induction hypothesis to each proper subgroup 5’ of H , the desired assertion follows by the transitivity of induced characters. (ii) Setting a = a ( H ) , we have

3 Rational valued characters

791

where, by convention, a(z-'gz) = 0 if z-lgz # H . Hence, by the definition of a , we have a H ( g )= 0 if g is not conjugate to a generator of H . Assume that g = y - l h y where h is a generator of H and y E G. Since aG is a class function, a G ( g ) = aG(h). On the other hand, z-*hz E H if and only if z E N G ( H ) since H =< h >. Thus, by (2),

as desired. W

Theorem 3.2. (Artin (1924), (1931)). Let x be a rational valued character of G . Then ( G ( xis an integral linear combination of characters of G induced from principal characters of cyclic subgroups of G.

Proof. Let H I , . . . ,H , be all nonconjugate cyclic subgroups of G and let hi be a generator of H i , 1 5 i 5 n. Consider the class function B on G defined by

By Lemma 3.1 (i) and transitivity of the induction, it suffices to show that B(g) = IGlx(g) for all g E G. Now fix g E G and choose a unique j E (1,. ..,n } such that < g > is conjugate to H j . Then, by Lemma 3.1 (ii), a(H;)G(g)= 0 for i # j , and Ct(Hj)G(g)= ING(Hj)l. Thus

But g and hj are Q-conjugate, so x ( g ) = ~ ( h j(by ) the proof of Lemma 17.5.1 (iv)) and therefore B ( g ) = IGlx(g). In what follows, we denote by R(G) the Q -space of all functions from G to Q which are constant on the Q -conjugacy classes of G.

Corollary 3.3. Let H I , . ..,H , be all nonconjugate cyclic subgroups of G. Then (i) G has precisely r irreducible Q -characters, say X I , . . . ,x7. (ii) X I , . . . ,xr and ( l ~ , ). .~ ,, ( 1 ~ , . are ) ~ two Q -bases for R(G).

.

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792

By Lemma 19.3.4 (with F = Q ), it suffices to show that each 1 )G,. ~ .~. ,( 1 ~ ~ ) ' .If H is any cyclic subgroup of G, then H is conjugate to some H i , in which case ( 1 ~ =) ( ~ 1 ~ ~ The ) ~ desired . assertion is now a consequence of Theorem 3.2.

Proof.

Q -character

x of G is a Q -linear combination of (

The following lemma due to Turull (1982) will enable us to provide an interesting application of Corollary 3.3 (namely, Corollary 3.7).

For any X E R(G), let A* : G -+ Q be defined by

Lemma 3.4.

X*(g) = 1/1 < g > I Then the map X

I+

X(x) x€

for all g E G

A* is an automorphism of the Q -space R(G).

Proof. It is clear that X H A* is a Q -linear map of R ( G ) into R ( G ) . Let G be a counter-example of minimal order. Then there exists X E R(G) and g E G such that X(g) # 0 and A* = 0. For any subgroup H of G, we have AH E R ( H ) and (AH)* = (X*)H = 0. Hence, by the choice of G , AH = 0 for every proper subgroup H of G. Thus G =< g > and X(z) = 0 if x E G and G #< x >. But if x E G and < g >=< z >, then X(z) = X(g), so X*(g) # 0,a contradiction. Let F be a field and let V be an FG-module. Given g E G , we put

Cv(g) = {w

E VJgv =

v}

Then Cv(g) is the maximal F < g >-submodule of V on which < g > acts trivially. In what follows all FG-modules and (Q G-modules are assumed to be finitely generated.

Corollary 3.5. (Turulk (1982)). Let V and W be QG-modules such that dimQ (Cv(g)) = dim^ (Cw(g)) for all g E G Then V Z W .

Proof. Let xv and xw be the characters afforded by V and W, respecHence, tively. Then xv, xw E R(G) and, by hypothesis, (xv)* = (xw)*. by Lemma 3.4, x v = xw and so, by Corollary 17.1.8 (ii), V % W.

3 Rational valued characters

793

Corollary 3.6. (Turu11 (1982)). Let F / Q be a finite Galois field extension and let V,W be FG-modules. If V is simple and

then

V E “W

for some u E Gal(F/(Q )

Proof. Let xv and x w be the characters afforded by V and W ,respectively. Then ~ ( x v=) C “(xv), X(xw) = C “ ( x w ) ,where u ranges over G a l ( F / Q ), are Q -characters of G. Hence X(xv), X(xw) E R(G). By hypothesis, X(xv)* = X(xw)* and so, by Lemma 3.4, X(xv) = X(xw). Since V is simple, we deduce that xv = “xw for some u E Gak(F/(Q),as we wished to show. W

.

Corollary 3.7. (Turu11 (1 982)). Let XI,.. ,X, be the conjugacy classes of cyclic subgmups of G, let Hj E X i and let aij be the number of double cosets HjgH; in G, 1 5 i , j 5 T . Then a;j is independent of the choice of

representatives H ; in X; and

(ajj)

is a nonsingular symmetric matrix.

Proof. Owing to Corollary 3.3 (ii), { ( 1 ~ , ) ~ 1I1i 5 r } is a Q -basis for R(G). By Mackey decomposition (Proposition 19.1.10), for any g E G , ((lH,)G)

= C ( l t H , t - ’ n < g > ) t

where t runs through a set of representatives of the double cosets < g > tH; in G. Hence [ ( l ~ , ) ~ ] * (isg )equal to the number of double cosets < g > t H ; in G. This shows that the a;j are independent of the choice of H ; in Xi. Furthermore, by Lemma 3.4,

is a Q-basis for R(G) and so ( a i j ) is nonsingular. Finally, since obviously aij = aji for each i , j , the result follows. W