The other <span class="mathjax-formula">$P(G,V)$</span>-theorem

The Other P(G; V )-Theorem

U. MEIERFRANKENFELD(*) - B. STELLMACHER(**)

Dedicated to Guido Zappa on his 90th birthday

1. Introduction.

Suppose thatHis a finite group,T2Sylp(H),pany prime, andVis an elementary abelian normalp-subgroup ofH. Then an elementary Frattini argument shows that eitherHˆC_H(V)N_H(J(T)) orJ(T)6C_H(V), where J(T) denotes the Thompson subgroup ofT.

In this paper we are interested in this second case. One of the questions is howJ(T):ˆJ(T)C_H(V)=C_H(V) is embedded inH:ˆH=C_H(V). The first problem is to find suitable properties of the Thompson subgroupJ(T) that can be expressed in terms ofHonly. This is done in the following way.

Recall thatJ(T) is generated by the elementary abelian subgroupsAof maximal order of T. It is evident that BCV(B) is elementary abelian for every subgroupBA. Hence by the maximality ofjAj

jAj jBCV(B)j ˆ jBkCV(B)jjV\Bj ¹ jBkCV(B)kCV(A)j ¹: This gives rise to the condition

jBkC_V(B)j jAkC_V(A)jfor every BA:

()

Note thatB:ˆC_A(V) yields an important special case of ():

jV=CV(A)j jA=CA(V)j:

( )

Both conditions () and ( ) can be phrased in terms of the factor groupH and theGF(p)H-moduleV just by replacing Aby its imageAinH. Evi- dentlyAsatisfies () with respect toV andHiffAsatisfies () with re- spect toV andH.

(*) Indirizzo dell'A.: Department of Mathematics Michigan State University East Lansing, MI 48824 USA

(**) Indirizzo dell'A.: Matematisches Seminar - Christian Albrechts UniversitaÈt Ludewig Meyn Strasse 4, D-24098 - Kiel - Germania

This consideration gives rise to the following definition in the more general set up of a finite groupGand a finite dimensionalGF(p)G-moduleV.

DEFINITION1.1. Let A be a subgroup of G such that A=C_A(V) is an elementary abelian p-group. Then A is an offender of G on V ifjV=C_V(A)j jA=C_A(V)j; and A is a best offender of G on V ifjBkC_V(B)j jAkC_V(A)jfor every BA. The normal subgroup of G generated by the best offenders of G on V is denoted by J_G(V).

In the literature, at least in the case of a faithful GF(p)G-module V, the set of best offenders is denoted byP(G;V).

The classical result aboutJ_G(V) is the following:

The PPP(G;V)-Theorem. Suppose that V is a faithful finite dimen- sional GF(p)G-module and that K is a component of G. Then either [J_G(V);K]ˆ1 or K/ J_G(V).

This theorem was proved by Timmesfeld [Ti] for the casepˆ2. Later Chermak [Ch] gave a proof for arbitraryp. Earlier proofs by Aschbacher [As] and Thompson (unpublished) used aK-group hypothesis.

As we also want to allow certain solvable analogues of components, we need one further definition.

DEFINITION1.2. A non-trivial subgroup K of J_G(V)is a J_G(V)-com- ponent if K is minimal with respect to Kˆ[K;J_G(V)].

The OtherPPP(G;V)-Theorem. Suppose that V is a faithful finite di- mensional GF(p)G-module and that Op(J_G(V))ˆ1. Then[E;K]ˆ1and [V;E;K]ˆ0for any two distinct J_G(V)-components E and K.

In the case of a faithful GF(p)G-module with O_p(J_G(V))ˆ1 we also show that every component of JG(V) is a JG(V)-component (see 3.4) and that every J_G(V)-component, which is not a component, is isomorphic to SL2(p)⁰,pˆ2 or 3 (see 3.2).

2. Offenders.

In this section Gis a finite group, p is a prime, andV is a finite di- mensionalGF(p)G-module. Some of the arguments in this section got their inspiration from [CD].

DEFINITION2.3. Let A be a subgroup of G.

(a) A actsquadraticallyon V if[V;A;A]ˆ0.

(b) j_A(V):ˆjAjjCV(A)j jVkC_A(V)j.

(c) V is a simple GF(p)G-module, if V6ˆ0and V and0are the only G- submodules in V.

LEMMA2.4. Let AG such that A=C_A(V)is elementary abelian. Then the following hold:

(a) j_A(V)ˆjA=CA(V)j jV=C_V(A)j.

(b) A is an offender on V iff j_A(V)1.

(c) If[V;A]ˆ0, then j_A(V)ˆ1.

(d) A is a best offender iff j_B(V)j_A(V)for all BA.

PROOF. (a) is obvious, and (b) and (c) follow from (a).

(d) By (a)jB(V)ˆj_BCA(V)(V) and so we may assumeC_A(V)B. Then C_A(V)ˆC_B(V) and soj_B(V)j_A(V) iffjBkC_V(B)j jAkC_V(A)j. p LEMMA2.5. Let AG be a best offender on V and BG be an offender on V. Then the following hold:

(a) A is an offender on V.

(b) B contains a best offender B on V with jB(V)jB(V) such that [V;B]6ˆ0or BˆB.

(c) A is a best offender on every A-submodule of V.

PROOF. (a): By 2.4(c), (d) 1ˆj1(V)jA(V), so by 2.4(b) A is an of- fender onV.

(b): ChooseBBsuch that firstj_B(V) is maximal and thenjBjis maximal. If [V;B]ˆ0, thenj_B(V)ˆ1 and thus by 2.4(b) alsoj_B(V)ˆ1.

Now the maximal choice ofjBjyieldsBˆB.

(c): LetWbe anA-submodule ofV andA0A. Then jA0kCW(A0)‡CV(A)j jA0kCV(A0)j jAkCV(A)j and thus

jA0kCW(A0)kCV(A)kCW(A0)\CV(A)j ¹ jAkCV(A)j:

SinceC_W(A₀)\C_V(A)ˆC_W(A) we getjA₀kC_W(A₀)j jAkC_W(A)j. p

LEMMA2.6. Let A and B be subgroups of G. Then j_hA;Bi(V)j_A\B(V)j_A(V)jB(V)

with equality iff CV(A\B)ˆCV(A)‡CV(B)andhA;Bi ˆAB.

PROOF. We may assume thatC_G(V)ˆ1 sincej_A(V)ˆj_ACH(V)=CH(V)(V), sojVjj_A(V)ˆ jAkC_V(A)j. Observe that

jhA;Bij jABjandjC_V(A\B)j jC_V(A)‡C_V(B)j:

Then

jVj²j_hA;Bi(V)j_A\B(V) ˆ jhA;BikC_V(hA;Bi)kA\BkC_V(A\B)j jABkCV(A)\CV(B)kA\BkCV(A)‡CV(B)j jAkBkC_V(A)kC_V(B)j

ˆ jVj²j_A(V)j_B(V):

p Fundamental for the investigation of best offenders is a replacement property first proved by Thompson, the Thompson Replacement Theorem, and then generalized by Timmesfeld, the Timmesfeld Replacement The- orem. We will use the following version of the Timmesfeld Replacement Theorem with [KS] as a reference.

LEMMA2.7. Let A be a best offender of G on V and W be a subgroup of V.

Then for A:ˆCA([W;A])the following hold:

(a) A is a best offender on V with j_A(V)ˆj_A(V).

(b) C_V(A)ˆ[W;A]‡C_V(A).

(c) [W;A]6ˆ0if[W;A]6ˆ0.

(d) [W;A;A]ˆ0.

PROOF. Properties (a) and (b) can be found in [KS, 9.2.1 and 9.2.3], (c) is stated there differently, so we give a proof here.

Assume that [W;A]ˆ0. Then by (b) W[W;A]‡C_V(A); in parti- cular [W;A]ˆ[W;A;A]. AsA=C_A(V) is ap-group, this last property gives [W;A]ˆ0.

Finally, by definition [W;A;A]ˆ0, and [A;A;W]ˆ0 sinceA=CA(V) is abelian. Hence the Three Subgroups Lemma implies (d). p

LEMMA2.8. Let A and B be quadratic offenders of G on V such that [A;B]C_G(V) andAC_G(V)\BC_G(V)ˆC_G(V):

ThenhA;Biis a quadratic best offender on V, or there exists a quadratic best offender X hA;Biwith

j_X(V)>maxfj_A(V);j_B(V)g:

PROOF. Let D:ˆ hA;Bi. ThenD=C_D(V) is an elementary abelian p- group. We may assume thatGis faithful onV, soDis elementary abelian andA\Bˆ1. From 2.6 we get thatjD(V)jA(V)jB(V) sincejA\B(V)ˆ1.

Assume first thatjD(V)>maxfj_A(V);jB(V)g. ThenjD(V)>1, and by 2.5(b) there exists a best offender DD with j_D(V)j_D(V)>1; in particular [V;D]6ˆ0. Now 2.7 gives the desired quadratic best offender XD.

Assume now thatjD(V)maxfjA(V);jB(V)g. By 2.6 jD(V)ˆjD(V)j1(D)j_A(V)jB(V);

so j_D(V)ˆj_A(V)j_B(V) and again by 2.6C_V(A)‡C_V(B)ˆC_V(A\B)ˆV.

This yields [V;A]ˆ[C_V(B);A] and [V;A;B]ˆ0, and the quadratic action

ofABfollows. p

LEMMA 2.9. Let AG and LG with Lˆ[L;A]. Then CA(L)\

\C_A([V;L;A])C_A([V;L]).

PROOF. Note thatO^p(L)ˆL hA^LisinceLˆ[L;A]. Thus, we may assume that V ˆ[V;L], in particular [V;A;A₀]ˆ0. Let A₀:ˆC_A(L)\

\C_A([V;L;A]). Then

[V;L;A0][V;hA^Li;A0]ˆ h[V;A;A0]^Li ˆ0: p

LEMMA2.10. Let A be a best offender of G on V, L/G, and W a simple L-submodule of V. Suppose that[L;A]6C_L(W). Then W is A-invariant.

PROOF. Note that [L;A]6C_G(W) implies [W;A]6ˆ0. Hence by 2.7 there exists a quadratic best offenderAAsuch that [W;A]6ˆ0 and [W;A;A]ˆ0. Then

[W;N_A(W)]W\C_V(A)W\W^b for everyb2A:

SinceW\W^bis anL-module andWis simple, we conclude that eitherWis A-invariant or [W;NA(W)]ˆ0. In the first case we are done, so we may assume that [W;N_A(W)]ˆ0. In particular

W\W^aˆ0 for everya2AnC_A(W):

(1)

Pickc2AnCA(W). Then

U:ˆWW^c ˆW[W;c]ˆW^c[W;c]:

(2) and

[W;c]U\C_V(A)U\U^bfor every b2A:

(3)

Assume that [W;c] is L-invariant. Then [W;c;A]ˆ0 implies that also [W;c;[L;A]]ˆ0. The decomposition (2) shows that [W;[L;A]]ˆ0, a con- tradiction. Thus we have

[W;c] is notL-invariant.

(4)

Letb2A. SinceU\U^bis anL-module andW is simple, we get from (3) that eitherU\U^bˆ[W;c] orUˆU^b. The first case contradicts (4), soU isA-invariant. But then, as [W;A;c]ˆ[W;c;A]ˆ0, we get that

CU(A)ˆCU(c)ˆ[W;c] andjU=CU(A)j ˆ jWj:

By 2.5(c)A is a best offender onU, sojWj ˆ jU=C_U(A)j jA=C_A(U)j.

Observe that jU^]j ˆ jWj² 1ˆ(jWj ‡1)jW^]j. On the other hand, by (1) any two distinctA-conjugates ofWintersect trivially, so there are at most jWj ‡1 such conjugates. We conclude thatjU=CU(A)j ˆ jWj ˆ jAj, and theA-conjugates ofWtogether with [W;c] form a partition ofU. Since all these conjugates of W are L-invariant, also [W;c] is L-invariant. This

contradicts (4). p

3. J_G(V)-Components.

As in the last sectionGis a finite group andV is a finite dimensional GF(p)G-module.

LEMMA 3.1. Suppose that V is a faithful GF(p)G-module. Let L/JG(V) with Op(L)ˆ1 and[L;JG(V)]6ˆ1. Then Lcontains a JG(V)- component of G.

PROOF. LetJ:ˆJ_G(V), and letK/Jbe minimal with respect toKL and [K;J]6ˆ1. It suffices to show thatKˆ[K;J].

Assume thatK6ˆ[K;J]. Then the minimality of K gives [K;J;J]ˆ1 and thus [K;K;K]ˆ1. Hence K is nilpotent. Again the minimality ofK

shows thatKis anr-group,ra prime. Moreoverr6ˆpsinceOp(L)ˆ1. As J is generated by p-elements we get that JˆO^r(J). Thus, [K;J;J]ˆ1

implies [K;J]ˆ1, a contradiction. p

In the next lemma we will use a result of Glauberman [Gl] as it is stated in [KS].

LEMMA 3.2. Suppose that V is a faithful GF(p)G-module. Let R be a J_G(V)-component with Op(R)ˆ1that is not a component of J_G(V). Then the following hold:

(a) pˆ2or3.

(b) RSL2(p)⁰andj[V;R]j ˆp².

(c) For every J_G(V)-component K with K6ˆR, [R;K]ˆ1 and [V;R;K]ˆ0.

PROOF. SetJ:ˆJ_G(V) andJ:ˆJ=Z(J). By 3.1R~ is a minimal normal subgroup ofJ. Hence, either~ R~is the product of components ofJ~orR~ is an elementary abelianq-group.

Assume first thatR~ is the product of components. Then an elementary argument shows thatR⁰is the product of components ofJ(see [KS, 6.5.1]).

By theP(G;V)-Theorem each of these components is normal inJ, so by 3.1 Ris a component, which is not the case.

Assume now thatR~ is aq-group. ThenRis nilpotent and thus alsoR is a q-group. Moreover q6ˆp since Op(R)ˆ1, so ROp⁰(G). Pick T₀2Syl_p(C_J(R)), and set

J0:ˆJ_NJ(T0)(V); W :ˆCV(T0) andJ0:ˆJ0=CJ0(W):

Observe thatJˆJ0C_G(R), soRis also aJ_J0(V)-component. By thePQ- LemmaRacts faithfully onW. AlsoC_J0(R)=J₀\T₀is ap⁰-group. LetXbe the inverse image of C_J0(R) inJ₀. Then [R;X]C_R(W), and the faithful action ofRgivesXC_J0(R), soXis ap⁰-group.

We have shown thatC_J0(Op⁰(J0))C_J0(R)ˆXOp⁰(J0). In addition, by 2.5(c)J0is generated by best offenders onW. HenceWandJ0satisfy the hypothesis of Theorem 9.3.7 in [KS]. It follows that either

pˆ2; RC₃ andj[W;R]j ˆ4; or pˆ3; RQ₈ andj[W;R]j ˆ9:

LetAJ₀be a best offender onVsuch that [R;A]6ˆ1. ClearlyRˆ[R;A]

andjA=CA(R)j ˆp. Moreover, according to 2.7 there exists a best offender AAsuch that [V;R;A]CV(A) and [V;R;A]6ˆ1. Hence 2.9 shows

that [R;A]6ˆ1, and we may assume thatAacts quadratically on [V;R].

But then again 2.9 shows thatCA(R)ˆCA([V;R]) andjA=CA([V;R])j ˆp.

AsAis an offender on [V;R] by 2.5(c), we getj[V;R]=C_[V;R](A)j ˆp. This gives [V;R]ˆ[W;R], and (a) and (b) hold.

Now letKbe any otherJ_G(V)-component. As we have seen aboveKis either a component or has a structure asR. In the first case the fact that GL2(p) is solvable forp3 shows that [V;R;K]ˆ0 and so also [R;K]ˆ1.

In the second case we can choose a best offenderBwithKˆ[K;B] such that hA;Bi is a p-group. Then C_hA;Bi(RK) is a normal p-subgroup of RKhA;Biand so centralizes [V;RK]. Hence the above result from [KS]

applies toRKhA;Bi=C_RKhA;Bi([V;RK]). Then [R;K]CR\K([V;RK])ˆ1

and [V;R;K]ˆ0. Hence (c) also holds in this case. p LEMMA3.3. Suppose that V is a faithful GF(p)G-module and that K is a J_G(V)-component with O_p(K)ˆ1. Then there exists a best offender A of G such that[K;A]ˆK and A is quadratic on[V;K].

PROOF. By 3.1 there exists a best offender B such that [K;B]6ˆ1.

Hence 3.2 gives [K;B]ˆK and thus [V;K;B]6ˆ0. Now 2.7 with W:ˆ[V;K] gives a best offender AB satisfying [W;B;A]ˆ0 and [W;A]6ˆ0. The first property shows thatAis quadratic onW. It remains to proveKˆ[K;A].

Assume that [K;A]6ˆK. Then again 3.2 yields [K;A]ˆ1 and thus by

2.9Wˆ[W;K]C_W(A), a contradiction. p

LEMMA3.4. Suppose that V is a faithful GF(p)G-module and that K is a component of J_G(V)with O_p(K)ˆ1. Then K is a J_G(V)-component.

PROOF. LetJ:ˆJG(V). By theP(G;V)-TheoremKis normal inJ, so Kˆ[K;J]. Hence, eitherKis aJ_G(V)-component, or there exists aJ_G(V)- component R<K. Suppose that the second case holds. Then RZ(K) sinceK is a component. ThusR is aJ_G(V)-component that is not a com- ponent. Hence 3.2 implies thatj[V;R]j ˆp². But thenGL([V;R]) is solvable and as above [V;R;K]ˆ0; particular [V;R;R]ˆ0. This is impossible since

Ris a non-trivialp⁰-group. p

LEMMA 3.5. Suppose that V be a faithful GF(p)G-module and Op(JG(V))ˆ1. Let K and L be two distinct components of JG(V). Then

[V;K;L]ˆ[V;L;K]ˆ0:

PROOF. We apply 3.4 and 3.5. Then there exist best offendersAandBsuch that [K;A]ˆKand [L;B]ˆL, andAandBact quadratically on [V;K] and [V;L], respectively. Moreover sinceKandLare normal inJ_G(V), we may assume thathA;Biis ap-group.

By way of contradiction we assume that [V;K;L]6ˆ0. Set G₀:ˆKLhA;Bi. Then there exists aG₀-submoduleWofVthat is minimal with respect to [W;K;L]6ˆ0. Since [K;L]ˆ1 also [W;L;K]6ˆ0, and the situation is symmetric inKandL.

Suppose that W6ˆ[W;K]. Then the minimality of W gives [W;K;K;L]ˆ0. On the other hand [W;K]ˆ[W;K;K], so [W;K;L]ˆ0, a contradiction. Hence we have Wˆ[W;K] and by symmetry also W ˆ[W;L]. ThusAandBare quadratic onW, and by 2.5(c)AandBare quadratic best offenders onW.

Let U be a maximal GF(p)G₀-submodule of W and set W^ :ˆW=U.

ThenW^ is a simpleGF(p)G₀-module. By 2.9 [W;C_A(K)]ˆ0 and similarly [W;C_B(L)]ˆ0. Thus we have

C_A(W)^ ˆC_A(K)ˆC_A(W) andC_B(W)^ ˆC_B(L)ˆC_B(W):

()

SinceAandBare quadratic best offenders onW, () shows thatAandBare also quadratic offenders onW. Hence there exist quadratic best offenders^ A^andB^onW^ inhA;Bisuch that [K;A]^ ˆKand [L;B]^ ˆL. In addition, we chooseA^ such thatjA^(W) is maximal with that property.^

Let X^ be a simpleK-submodule ofW. Assume that [K;^ B]^ 6ˆ1. Then 2.10 shows thatX^is normalized by everyL-conjugate ofB, so^ Lnormalizes X. As^ X^ is a simpleK-module, Schur's Lemma shows thatEnd_K(X) is a^ finite division ring and then Wedderburn's Theorem thatL=CL(X) is cyclic.^ This shows that [X;^ L]ˆ0 sinceLis perfect. HenceCW^(L) is a non-trivial G₀-submodule of W. As^ W^ is a simple G₀-module, we conclude that [W;^ L]ˆ0, a contradiction.

We have shown that [K;B]^ ˆ1 and similarly [L;A]^ ˆ1. Observe that C_hA;Bi(KL) is a normalp-subgroup ofG0and hence centralizesW, so^

[A;^ B]^ C_G0(W) and^ AC^ _G0(W)^ \BC^ _G0(W)^ ˆC_G0(W):^

Now 2.8 and the maximality ofj_A^(W) show that^ hA;^ Bi^ is a quadratic best offender onW. But the above argument with^ hA;^ Bi^ in place ofB^ implies that [K;hA;^ Bi]^ ˆ1, which contradicts [K;A]^ ˆK. p The proof of The Other P(G;V)-Theorem. Let E andK be distinct J_G(V)-components. Then by 3.2(b) and 3.6 [V;E;K]ˆ0ˆ[V;K;E]. So by the Three Subgroups Lemma, [E;K]ˆ1.

REFERENCES

[As] M. ASCHBACHER,GF(2)-Representations of finite groups, Am. J. Math.104 (1982), pp. 683-771.

[Ch] A. CHERMAK, Quadratic action and the PG(V)-theorem in arbitrary characteristic, J. Group Theory2(1999), pp. 1-13.

[CD] A. CHERMAK, A. DELGADO,A measuring argument for finite groups, Proc.

AMS107(1989), pp. 907-914.

[Gl] G. GLAUBERMAN,Failure of factorization in p-solvable groups, Quart. J.

Math. Oxford24(1973), pp. 71-77.

[KS] H. KURZWEIL, B. STELLMACHER, Theorie der endlichen Gruppen. Eine EinfuÈhrung.Springer-Verlag, Berlin, 1998, p. 341.

[Ti] F. TIMMESFELD, A remark on Thompson's replacement theorem and a consequence, Arch. Math.38(1982), pp. 491-499.

Manoscritto pervenuto in redazione il 31 agosto 2005.