On the Structure of Induced Modules and Tensor Induction for Group Representations (*).
EMANUELEPACIFICI(**)
Dedicated to Professor Guido Zappa on the occasion of his 90th birthday
ABSTRACT- LetVbe a simple module for a finite groupG, over a finite fieldF, and let H be a subgroup ofG. Assuming thatV is induced by anFH-module, we in- vestigate some aspects of the structure ofVviewed as a module forH. This kind of analysis turns out to play a central role in a problem concerning tensor in- duction for representations of finite groups.
Introduction.
I.LetGbe a finite group,Fa finite field, andV a simpleFG-module;
given a subgroupHhaving odd index inG, and anFH-submoduleWofV, assume thatV is isomorphic to theinduced module W"G. In this setting, we are interested in exploring the structure of theFH-moduleV#H(which is V restricted to H) from a particular point of view: namely, we ask whether the odd-index assumption forHimplies thatthe multiplicity of W as a composition factor in the socle of V#His also odd. By Lemma 1.4(b), this is equivalent to saying that V#H is isomorphic to the FH-module Ls
i1W
Y, wheresis an odd positive integer, andY is a submodule of V#H not containing any submodule isomorphic toW.
It follows from Clifford's Theorem ([1, 11.1]) that the answer to the above question is certainly affirmative whenHis anormalsubgroup ofG.
More generally, as outlined in the last paragraph of Section 2, it is not difficult to see that the same holds whenWis induced from the normal core
(*) 2000Mathematics Subject Classification. Primary: 20C15, 20C20, 20C25.
(**) Indirizzo dell'A.: Dipartimento di Matematica ``Federigo Enriques'', Uni- versitaÁ degli Studi di Milano, Via C. Saldini 50, 20133 Milano, Italy.
LofHinG,provided L has odd index in H(hence inG). On the other hand, ifWis induced fromLbutjH:Ljis even, then the answer can be negative, as itis shown by an example ([5, 11.1]) in whichGis solvable,Fis the prime field in characteristic 3, andjG:Hjis 3.
In view of our original motivation for this kind of analysis (presented in Part II of this Introduction), we are actually interested in the case whenW isnotinduced fromL, and we can also assume thatFhas odd characteristic.
The main result of this paper (which is proved in Section 2) is the following.
THEOREMA. Let G be a finite solvable group, H a subgroup of G having odd index, F a (not necessarily finite) field of odd characteristic, V a simpleFG-module, and W a submodule of V#Hsuch that V'W"G. De- noting by L the normal core of H in G, assume that W is not induced from L, and that G=Lis a Frobenius group with Frobenius complementH=L.
Then we have V#H' Ls
i1W
Y, where s is an odd number and Y is a submodule of V#Hsuch thatHomFH(W;Y)0 (In other words,Whas odd multiplicity as a composition factor in soc(V#H)).
Note that, if His a (notnormal) subgroup of oddprimeindex in the solvable groupG, thenG=Ldoes have the structure of a Frobenius group with Frobenius complement H=L (in fact, denoting by K=L a minimal normal subgroup ofG=L, we have thatG=Lis a semidirectproductofK=L and H=L; moreover, every nontrivial element of H=L acts fixed-point- freely by conjugation onK=L). Therefore Theorem A covers this case, thus providing a generalization of Theorem 9.7 in [5].
II.It may be worth mentioning the problem which led us to the question presented in Part I.
Let G be a finite group, H a subgroup of G, and D an irreducible complex representation forG. As itis easy to check, every directsummand of the restrictionD#Hmusthave degree atleastas large as degDdivided by the indexjG:Hj, andDis induced by a representation ofHif and only if D#H does have a directsummand of degree (degD)=jG:Hj. One of the main purposes of [5] is to explore the possibility of an analogous result for tensor induction; more explicitly we ask whether the following holds.
CONJECTURE. Let D be a faithful, quasi-primitive and tensor-inde- composable representation of G. Then D is tensor-induced by a projective representation of H if and only if D#Hhas a tensor factor whose degree is thejG:Hjth root ofdegD.
(We refer to [5, Introduction and Section 1] for a detailed discussion about the concept of tensor induction, which motivates and explains the setting of the above Conjecture.) Following the line developed in [5], this problem can be approached by means of two subsequent reductions. First, the Conjecture appears to be deeply linked to a statement ([5, `weak' Conjecture 4.3]) concerning form inductionfor symplectic modules over finite fields (see Section 3), and at this level it can be shown that the Conjecture is false in its full generality (Example 5.2 in [5]). Next, positive results toward the Conjecture are obtained assuming thatHhas odd index in G, and such results are achieved via a reduction to the question pre- sented in Part I. In particular, the Conjecture is proved to be true whenH is a normal subgroup of odd index inG(provided the Fitting subgroup ofG is assumed noncentral) and also, through Theorems 9.7 and 9.10 of [5], whenGis solvable andHhas oddprimeindex inG.
As in this paper we generalize [5, 9.7 and 9.10] (by means of Theorem A and Theorem 3.3 respectively), we are in a position to extend the cases in which the Conjecture (together with the weak version of Conjecture 4.3 in [5]) is proved to be true. The precise statements for these results, together with the relevant definitions and notation, are formulated in Section 3.
To conclude, every abstract group considered throughout the following discussion is tacitly assumed to be finite. Also, we shall freely use (often with no reference) some basic facts in Representation Theory, such as Clifford's Theorem, Mackey's Lemma ([1, 10.13]) and Nakayama re- ciprocity ([3, VII, 4.5 and 4.10]).
1. Some preliminaries.
Before proving Theorem A, we recall some results and notation which will be relevantin the sequel.
LEMMA1.1. Let G be a Frobenius group with Frobenius complement H and Frobenius kernel K, and let I be a subgroup of G such that I\H61.
Then we have I(I\H)(I\K).
PROOF. See [4, 4.1.8]. p
LEMMA 1.2. Let H be a solvable Frobenius complement of even order, which does not have any subgroup of index2. Then there exists a normal subgroup N of H such that H=N is isomorphic to the alternating group A4.
PROOF. SetA:O2(H), andN:CH(A); looking atthe proof of Zas- senhaus' Theorem 18.2 in [6], we see that our assumptions force At o be isomorphic to the quaternion group of order 8, whence H=N embeds in Aut(Q8)'S4. Moreover, a Sylow 2-subgroup ofH=Nmustbe isomorphic toC2C2. AsH=Ncan notbe a 2-group, its order is necessarily divisible
by 3, and the claim follows. p
LEMMA1.3. Let H be a group,Fa field, and M a normal subgroup of H.
Also, let W be a simple FH-module, and U a simple constituent of W#M. If I is the inertia subgroup of U in H, and e denotes the multiplicity of U as a composition factor in W#M, thenjI=Mj e2(dimFEndFM(U))=
(dimFEndFH(W))holds.
PROOF. Letfdenote the multiplicity ofWas a composition factor of the largestsemisimple quotientofU"H. Since the direct sum off copies ofWis a homomorphic image ofU"H, the direct sum ofef copies ofUis a homo- morphic image ofU"H#M. From Mackey's Lemma, it is easy to see that U"H#Mis semisimple and one of its homogeneous components is the direct sum ofjI=Mjcopies ofU: thereforejI=Mj ef. By [3, VII, 4.13],
edimFEndFM(U)fdimFEndFH(W):
Thus
jI=Mj ef e2(dimFEndFM(U))=(dimFEndFH(W));
as claimed. p
LEMMA1.4. Let G be a group, H a subgroup of G,Fa field, V a simple FG-module, and W a submodule of V#Hsuch that V'W"G. Let T be the homogeneous component of W in the socle of V#H. Then the following conclusions hold:
(a) the multiplicity of W as a composition factor in T is given by (dimFEndFG(V))=(dimFEndFH(W));
(b) T is a direct summand in V#H, it has a unique direct comple- ment Y, and Y is such thatHomFH(W;Y)HomFH(Y;W)0.
PROOF. Claim (a) easily follows from the fact that, by [3, VII, 4.12b)], the multiplicity ofWas a composition factor inTis given by
(dimFHomFH(W;V#H))=(dimFEndFH(W)):
Nakayama reciprocity yields now the conclusion.
For claim (b), note that if Zis any submodule ofV#H with the same dimension as W, t hen Z is a directsummand (simply becauseV is the vector space direct sum of the translates of Z, the translates different fromZare permuted byHamong themselves, and so their sum is anH- module complementto Z). Let Y be of minimal dimension among the submodules ofV#Hsuch thatTYV#H. IfYcontained a submoduleZ isomorphic toW, thenZwould lie inTandZwould be a directsummand of Y, cont rary t o t he minimalit y of Y. Therefore we musthave HomFH(W;Y)0, andT\Y 0 (so t hat Yis a directcomplementtoT).
Dually, one can use the fact that ifZ0is any submodule of V#Hwithco- dimensionequal to dimWthen itis a directsummand: ifYhad a nonzero homomorphism ontoW, t he sum of Twith the kernel of that could play the role ofZ0and yield a contradiction. Thus HomFH(Y;W)0, and from this itfollows atonce thatthere can be no directcomplementtoTother
thanY. p
LEMMA1.5. Let H be a group, L a normal subgroup of H,Fa finite field, and S a1-dimensionalFH-module whose kernel contains L. Let W be a simpleFH-module. Then WS and W have the same (nonzero) multi- plicity as composition factors in the socle of W#L"H.
PROOF. See [5, 9.1]. p
LEMMA1.6. Let H be a group, L a normal subgroup of H,Fa finite field, and W an absolutely simpleFH-module. Assume that there exists anFH- module S such that kerS contains L,jH:kerSj 2, and WS is iso- morphic to W. Then the multiplicity of W as a composition factor in the socle of W#L"His an even (positive) number.
PROOF. See [5, 9.4]. p
REMARK1.7. It is not hard to see that the ideas of the proof of [5, 9.7]
can be applied more generally, and we shall need some of their con- sequences here. Let Gbe a (finite) group,Ha subgroup ofG,Fa finite field, V a simple FG-module, and W a submodule of V#H such that V 'W"G. Then EndFG(V) and EndFH(W) are fields, every elementof the latter arises as the restriction of one and only one element of the former, and the relevant elements of EndFG(V) form a subfield: call thatK, write VKforVregarded asKG-module, andWKforWregarded asKH-module (of courseVKis simple, and itis induced byWKfromH). Itis now easy to
see that EndKG(VK) EndFG(V), and EndKH(WK) EndFH(W), soWK
is indeed absolutely simple, and by Lemma 1.4(a) the multiplicity ofWas a composition factor in soc(V#H) is the same as the multiplicity ofWKas a composition factor in soc((VK)#H). Moreover, if WK is induced from some subgroup L of H, t hen WK has a submodule of K-dimension dimK(WK)=jH:Lj; that subspace is also a submodule of W, of F-di- mension dimF(W)=jH:Lj, and so W is also induced fromL.
2. A proof of the main theorem.
We present next a proof of Theorem A, which was stated in the In- troduction. We are interested in this result when the fieldFis finite (and this will be our assumption), but in Remark 2.1 we shall take the oppor- tunity to explain that the theorem is in fact true also ifFis infinite. It may be worth stressing that in the special case whenFis a splitting field forG (for instance, whenFis algebraically closed), Theorem A is an immediate consequence of Lemma 1.4(a).
PROOF OFTHEOREMA. In what follows, we shall assume the statement true for all groups having order strictly smaller thanjGj, and our aim will be to show that the statement is true forGas well. As the first step, we shall prove thatWcan be assumed absolutely simple.
In fact, let us suppose that Theorem A is true when the relevantH- module is absolutely simple. Taking in account Remark 1.7 and its set-up, we can apply Theorem A withK,VK andWK in place ofF,V andW re- spectively. Then we get that the multiplicity ofWKas a composition factor in soc((VK)#H) is odd. But, as explained in 1.7, that multiplicity equals the multiplicity ofW as a composition factor in soc(V#H), and we achieve the desired conclusion.
In view of the previous step, we henceforth assume thatWis absolutely simple. LetXbe a simple constituent ofW#L, and letIdenote the inertia subgroupIG(X) (recall that this is the subgroup of all the elementsgofG such thatXgis isomorphic toXas anFL-module). Also, denote byK=Lthe Frobenius kernel ofG=L. We shall proceed by discussing the various si- tuations which may occur, depending onI.
(1). Case I\HL.This can nothappen, as otherwise we would get IH(X)L, and Clifford's Theorem would yield that W is induced by X fromL, against the hypothesis.
(2). Case L<I\H<H. Set J:IK(I\H)K (see Lemma 1.1), and letUbe the (unique) submodule ofW#I\Hwith the property thatU#L is the homogeneous component of W#L containing X. Since U"H'W, this U mustbe absolutely simple. Moreover, we get(U"J)"G'U"G' '(U"H)"G'W"G'V, so t hat U"Jis a simpleFJ-module. Of courseJis a solvable group, I\H is a subgroup of ithaving odd index, L is the normal core ofI\HinJ, andJ=Lis a Frobenius group with Frobenius complement(I\H)=L. Moreover, U is notinduced from L. By our in- ductive hypothesis, we can conclude that U has odd multiplicity as a composition factor in the socle of (U"J)#I\H. By Lemma 1.4(a), this is equivalentto saying thatdimFEndFJ(U"J) is an odd number. Now, we have
dimFEndFG(V)dimFEndFJ(U"J)(dimFEndFG(V))=(dimFEndFJ(U"J));
and it suffices to show that U"J has odd multiplicity (as a composition factor) in soc(V#J). We shall see that this multiplicity is in fact 1.
Denoting byT a transversal forI\HinH, we get V#J'W#I\H"J' M
t2T
Ut
"J'U"J M
t2T n(I\H)
Ut
"J
(Observe that, although the individual Ut are notnecessarily F[I\H]- modules, certainly L
t2T n(I\H)Utis invariant under the action ofI\H). For a proof by contradiction, suppose that the multiplicity ofU"Jin soc(V#J) is greater than 1; this means that HomFJ
U"J; L
t2T n(I\H)Ut
"J which is isomorphic, as a vector space, to HomF[I\H]
U; L
t2T n(I\H)Ut
"J#I\H is notthe zero space. Therefore,Xis a constituent of L
t2T n(I\H)Ut
"J#Land, finally, there existtinT n(I\H) andjinJsuch thatXis a constituent of (Utj)#L. Now,X(tj) 1is a constituent ofU#L, so thattjlies inI. Writingjas hk, where h is in I\H and k in K, we have that thktj is in I(I\H)(I\K) (see Lemma 1.1). This implies thattlies inI\H, which is notthe case.
(3).Case IH.We see that in this situation W has multiplicity 1 in soc(V#H). In fact,W#Lis now a homogeneous componentofV#L. IfV#H contained another isomorphic copy ofW, the restriction of that toLwould be isomorphic to W#L; but this is a contradiction, as a homogeneous component can never be isomorphic to any submodule distinct from it.
(4). Case H<I<G.We have that I is a solvable group, H is a subgroup ofIhaving odd index,Lis the normal core ofHinI, andI=Lis a Frobenius group with Frobenius complementH=L. Also, letRbe the submodule ofV#Igenerated by the subspaceW(so thatRis isomorphic to W"I, and itis certainly a simple FI-module). By the inductive hy- pothesis we deduce that dimFEndFI(R) is an odd number and, as dimFEndFG(V) is given by that number times the multiplicity of R in soc(V#I), it is enough to show that the latter multiplicity is odd. But, similarly to what happens in Case (3),R#Lis a homogeneous component of V#L and, as above, there can not be any other copy of R in V#I: therefore the multiplicity ofRin soc(V#I) is 1, and the argument for this case is complete.
(5). Case IG.Our assumption thatG=Lis a Frobenius group with Frobenius complementH=Limplies that, considering the action of Hon the set of its right cosets inG(given by right multiplication), the orbits not containing the trivial coset H have a common length, namely jH:Lj.
Therefore, Mackey's Lemma applied to the present situation yields V#H'W Mn
i1
W#L"H
;
wherenis the number of nontrivial double cosets ofHinG. This number is given by (jG:Hj 1)=jH:Lj, so there is nothing to prove ifjH:Ljis odd (in that casenis even). From now on we shall then assumejH:Ljeven, and most of the time our aim will be to show thatWhas even multiplicity as a composition factor in soc(W#L"H).
Let us start by assuming thatHhas a subgroupQwhich containsLand is such thatjH:Qj 2; then we can consider the representation ofH=Q which maps the generator to 1 inF(and view it as a representation for H). We claim that, if S denotes an FH-module associated to this re- presentation, then WS is isomorphic to W. In fact, by Lemma 1.5, WSandWhave the same multiplicity (call itr) as composition factors in the socle ofW#L"H. If they are assumed to be nonisomorphic, then Lemma 1.4(a) yields
nr1 jEndFG(V): EndFH(W)j jEndFG(V): EndFH(WS)j nr (here we used that EndFH(W) and EndFH(WS) are isomorphic vector spaces), a clear contradiction. We are now in a position to apply Lemma 1.6 (as of course the kernel of Shas index 2 inH), and we are done in this case.
IfH=Ldoes not have a subgroup of index 2, then (by Lemma 1.2) there exists a normal subgroup N of H, containing L, such that H=N is iso- morphic toA4. In whatfollows, we denote byMthe subgroup ofHwhich containsNand such thatM=Nis the Sylow 2-subgroup ofH=N.
Letus assume thatW#M is nothomogeneous. Then we getW#M
UUhUh2, wherehis inHnMand the three summands are simple homogeneous components. Now,Mis a subgroup ofKMhaving odd index, L is the normal core ofMin KM, andKM=Lis a Frobenius group with Frobenius complementM=L. SinceWis induced byUfromM, we see that U is absolutely simple; moreover, U"KM is simple, as itinducesV. Now, M=Ldoes have a subgroup of index 2 and, sinceU"KM#Lis homogeneous, we can apply the same argument as in the first two paragraphs of Case (5) (withKM,M,U"KMandU in place ofG,H,V andW respectively) con- cluding that the multiplicity of U in soc(U#L"M) is even: say, 2k. Since IG, the restriction ofVtoLis homogeneous; thusU#LandUh#L, which are submodules of equal dimension inV#L, mustbe isomorphic. From this, we get
HomFM(Uh;U#L"M)'HomFL(Uh#L;U#L)'HomFL(U#L;U#L)' 'HomFM(U;U#L"M);
whence Uh and (similarly) Uh2 have the same multiplicity as U in soc(U#L"M). Now we have
W#L"M'(U#LUh#LUh2#L)"M'U#L"MU#L"MU#L"M' ' M6k
i1
U
M6k
i1
Uh
M6k
i1
Uh2
Z;
where the socle ofZdoes not contain any of theUhias a submodule (from each of the three copies ofU#L"Mwe ``extracted'' all the copies of eachUhi; note that every submodule ofU#L"Misomorphic to one of theUhiis cer- tainly a direct summand ofU#L"M). Finally,
W#L"H' M2l
i1
W Z"H;
(wherel:9k) with HomFH(W;Z"H)'HomFM(W#M;Z)0.
It remains to examine Case (5) in the situation in whichH=Ldoes not have a subgroup of index 2, andW#Mis homogeneous. Observe that in this caseW#Mis simple because, by Lemma 1.3, ifedenotes the multiplicity of a simple constituent ofW#Min it, we havee2 jH=Mj 3. The composition
length (as an F[KM]-module) of W#M"KM'V#KM can be 1, 2, or 3. We analyze the situation in each of the three cases.
Assume that the composition length of V#KM is 3, and set V#KM' 'Z1Z2Z3 where theZiare simpleF[KM]-modules. Since we get HomFM(W#M;Zi#M )'HomF[KM](W#M"KM;Zi)'HomF[KM](V#KM;Zi)'
'HomFG(V;Zi"G)' EndFG(V);
the multiplicity ofW#Min soc(Zi#M) is dimFEndFG(V)=dimFEndFM(W#M).
In particular, this number does not depend oni, so that the multiplicity of W#M in the socle ofV#M is a multiple of 3. On the other hand, Mackey's Lemma gives
V#M'W#M"KM#M'W#M Md
j1
W#L"M
;
where d is the number of double cosets of M in KMdifferentfrom M.
This number is given by (jKM:Mj 1)=jM:Lj 3(jG:Hj 1)=jH:Lj, whence the multiplicity of W#M in the socle of V#M is congruentto 1 modulo 3. We thus reached a contradiction, so this case can not arise.
Let us now examine the case in which the composition length ofV#KMis 2, so that we have dimV2kdimX. On the other hand, dimV is given by jG:HjdimWsjG:HjdimX, wheres denotes the composition length of W#L. The conclusion is thatsis even, say 2r; therefore we get
W#L"H'M2r
j1
(X"H);
and of course we are done.
Finally, letV#KMbe simple, and letSbe anFM-module such that kerS containsL, andjM:kerSj 2. IfW#MS6'W#M, then a contradiction arises as in the second paragraph of Case (5); therefore we must have W#MS'W#M. IfW#Mis absolutely simple, then we apply Lemma 1.6 getting that dimFHomFM(W#M;W#L"M) is an even number; we reach now the desired conclusion, as HomFM(W#M;W#L"M) is isomorphic to HomFH(W;W#L"H). We are left with the case in whichW#Mis notabso- lutely simple: in such a situation, the Theorem stated in the Introduction of [2] guarantees thatW#M"His isomorphic to a direct sum of three copies ofW, and also that EndFM(W#M) has degree 3 as a field extension ofF.
Observe that we can also assumeW#M notinduced fromL, otherwise the composition length ofW#L(as anFL-module) is the even numberjM:Lj,
and again we are done. Now, we get
HomFM(W#M;V#M)'HomFH(W;V#M"H)'HomFH(W;(V#KM)#M"H)' 'HomFH(W;V#KM"G#H)'HomFH(W;V#HV#HV#H);
where the last isomorphism relation holds because of the following:
V#KM"G'(W#M"KM)"G'W#M"G'(W#M"H)"G'(WWW)"G'VVV:
The conclusion so far is
dimFHomFM(W#M;V#M)=dimFEndFM(W#M)3dimFHomFH(W;V#H)=3
dimFHomFH(W;V#H);
in other words, the multiplicity ofWin the socle ofV#Hequals the multi- plicity ofW#Min the socle ofV#M. This completes the proof, as we can now use the inductive hypothesis and conclude that the latter multiplicity is odd.
p REMARK 2.1. Let G be a finite group, and F a field of prime char- acteristic. Denoting by n the order of G, we set F(n) to be the (finite) subfield of the algebraic closure ofFgenerated by then-th roots of 1, and we defineF0:F(n)\F. In this setting, Lemma 6 of [2] establishes what follows: for every subgroup X ofG, and for every simpleFX-module U, there exists a simple F0X-module U0, uniquely determined up to iso- morphisms, such thatU'U0F0F(we refer here to Definition 1.1b) of [3, VII]).
The above result enables us to prove Theorem A in its full generality, without requiring thatFis finite. In fact, assume the hypotheses of The- orem A as stated in the Introduction, and consider the modulesW0andV0 (anF0H-module and anF0G-module respectively) associated toW andV by means of [2, Lemma 6]. It is easy to check thatW0andV0satisfy the hypotheses of Theorem A (for this purpose, it is convenient to take in ac- count that the process of induction of modules ``commutes'' with the pro- cess of tensoring modules with a field extension); thus, as we proved Theorem A when the relevant field is finite, we can conclude that (dimF0EndF0G(V0))=(dimF0EndF0H(W0)) is an odd number (here we also applied Lemma 1.4). Now, using 1.12 and 1.1a) of [3, VII], we get
dimFEndFG(V)dimF( EndF0G(V0)F0F)dimF0EndF0G(V0) and, similarly, dimFEndFH(W)dimF0EndF0H(W0). Another appeal to 1.4 completes the argument.
We stress that, as mentioned in the Introduction, the assumption ofW notbeing induced fromLis crucial for Theorem A (see [5, 11.1]),although that assumption is not needed when jH=Lj is odd. In fact, assumingW induced from theFL-moduleX, itis easy to see (using [3, VII, 4.12b)] and Clifford's Theorem) that the multiplicity ofWas a composition factor in the socle ofV#His given byjIG(X):IH(X)j, a divisor of the odd numberjG=Lj.
3. Form induction and tensor induction.
We start this section recalling some definitions and notation. For fur- ther details, we refer to [5, Introduction, Section 1 and Section 3].
DEFINITION3.1. LetGbe a group,Fa field,V anFG-module, andf a symplectic F-form defined on (the underlying vector space of) V; if f(ug;vg)f(u;v) holds for allu,vinV andginG, thenf is calledG-in- variant.
DEFINITION3.2. LetGbe a group,Ha subgroup ofG,Fa field,V a simpleFG-module, andWa submodule ofV#H. Assume that aG-invariant nonsingular symplectic F-form f is defined onV, and that the following conditions hold:
(a) the restriction off toWW, which is anH-invariantsym- plecticF-form onW, is nonsingular;
(b) the translateWg lies inW? for allginGsuch thatWg6W;
(c) V is induced byW fromH.
Then we say thatV isform-inducedbyW (with respect tof) fromH.
A mapP:H!GL(d;F) is called a projective representation ofH(of degreed, over the fieldF) if the mapP, defined as the composite of Pwith the natural homomorphism ofGL(d;F) ontoPGL(d;F), is a group homo- morphism. If P1 andP2 are projective representations of Hhaving the same degreed, then they are called equivalent ifP2is the composite ofP1
with an inner automorphism ofPGL(d;F); in this case, we writeP1'P2. Given two projective representationsP andQof H, having degreesc anddrespectively, the symbolPQdenotes the inner tensor product ofP and Q (which is a projective representation of H whose degree is cd), whereas the symbolP"Gdenotes the projective representation ofGwhich is tensor induced byPfromH.
In order to achieve the desired results on form induction of modules, and consequently on tensor induction of representations, we need (to- gether with Theorem A) a generalization of Theorem 9.10 in [5]. This generalization is only stated in that paper, so we present next a proof.
THEOREM3.3. Let G be a solvable group, H a subgroup of G having odd index,Fa finite field, V a simpleFG-module which carries a G-invariant nonsingular symplectic F-form f , and W a submodule of V#H such that V 'W"G. Assume that W is induced from the normal core L of H in G.
Then there exists a submodule Z of V#Hsuch that f does not vanish on Z, V 'Z"G, and Z has odd multiplicity as a composition factor insoc(V#H).
PROOF. We proceed by induction onjG:Hj. IfHis a maximal subgroup ofG, then we get the conclusion applying [5, 9.10]; thus we shall assume that there exists a proper subgroupEofGsuch thatHis properly contained inE.
Now,Vis induced byWfromH, so we getV '(W"E)"G; denoting byRthe moduleW"E, we have thatVis induced byRfromE, andRis in turn induced from a normal subgroup ofGcontained inE(which isL). We conclude thatR is induced from the normal core ofEinGand, sincejG:Ejis odd, we can apply the inductive hypothesis (we can certainly assume that Ris a sub- module ofV#E) and find a submoduleSofV#Esuch thatfdoes notvanish on S,V 'S"G, andShas odd multiplicity as a composition factor in soc(V#E).
Next, we know that there exists a submodule X of V#L such that V 'X"G; by Mackey's Lemma we get
V#E'M
t2T
(Xt)"E
whereT is a set of representatives for the double cosets inGofLandE.
Since each of the (Xt)"EinducesVfromEand is therefore simple, we have thatSis isomorphic, as anFE-module, to one of those. We conclude thatS is induced fromL, hence also from the normal core ofHinE. Therefore we can use again the inductive hypothesis, obtaining that there exists a sub- moduleZofS#Hsuch thatf does notvanish onZ,S'Z"E, andZhas odd multiplicity as a composition factor in soc(S#H). Now, putting together the two steps, we see thatZsatisfies the required conditions. p We are now in a position to extend Theorem 10.1 and Theorem 10.2 of [5]. The two theorems below are only stated, as a proof of them can be obtained arguing as in 10.1 and 10.2 of [5], just replacing Theorem 9.7 and Theorem 9.10 of [5] with Theorem A and Theorem 3.3 of this paper.
THEOREM3.4. Let G be a solvable group, H a subgroup of G having odd index,Fa finite field, V a simpleFG-module, and W a submodule of V#H. Denoting by L the normal core of H in G, assume that G=L is a Frobenius group with Frobenius complement H=L. Assume also that V carries a G- invariant nonsingular symplecticF-form f which does not vanish on W. If V is induced by W from H, then V is also form-induced from H (with re- spect to f ).
THEOREM3.5. Let G be a solvable group, H a subgroup of G having odd index, and D a faithful, primitive, tensor-indecomposable representation of G. Denoting by L the normal core of H in G, assume that G=L is a Frobenius group with Frobenius complement H=L. Assume also that we haveD# H'P1P2, where P1and P2are projective representations of H.
If degP2 is not1, and (degP2)jG:Hj is a divisor ofdegD, then we have (degP2)jG:HjdegD, and there exists a projective representation P of H such thatD 'P"Gholds.
Acknowledgments. The author wishes to thank Professor C. Casolo and Professor L.G. KovaÂcs for their valuable advice.
REFERENCES
[1] C. W. CURTIS- I. Reiner,Methods of representation theory I, Wiley, New York, 1981.
[2] S. P. GLASBY- L.G. KOVAÂCS, Irreducible modules and normal subgroups of prime index, Commun. Algebra,24(1996), pp. 1529±1546.
[3] B. HUPPERT, N. BLACKBURN,Finite groups II, Springer, Berlin, 1982.
[4] H. KURZWEIL, B. STELLMACHER, The theory of finite groups, Springer, New York, 2004.
[5] E. PACIFICI, On tensor induction for representations of finite groups, J.
Algebra,288(2005), pp. 287±320.
[6] D. PASSMAN,Permutation groups, W.A. Benjamin Inc., New York, 1968.
Manoscritto pervenuto in redazione l'8 settembre 2005.
