Almost quasisimple groups

| {z }

tfactors

bySt. ClearlyG1 ≤GL(V1) is a p-solvable irreducible linear group. Thusb(G₁)≤t(q) and b^∗(G₁)≤t(q) by induction on the dimensionm of V1 and by Corollary 4.3.

First let q ≥ 5. Then t(q) = 2. Thus b(G) ≤ 2 follows from [45, Theorem 3.6]

unless (m, t) = (2,2). In case (m, t) = (2,2), that is, G ≤ G1 o_cS2 ≤ GL4(q) for some p-solvable group G₁ ≤ GL₂(q) let x₁, y₁ ∈ V₁ be a basis of V₁ satisfying either NG1(hx₁i)⊆NG1(hy₁i) or the property that every non-identity element ofCG1(x1) takes y1 toy1+x1. (Such a basis exists by Theorem 4.1.) We claim that ifα∈Fq\ {0,1}then x₁⊗x₁, y₁⊗(y₁+αx₁) is a base forG₁o_cS₂ ≥G. Indeed, letg= (A⊗B)σ∈G₁o_cS₂ with A, B ∈ G1, σ ∈ S2 fixing these two vectors. Then g(x1 ⊗x1) = x1⊗x1 implies that Ax1 = λx1, Bx1 = λ⁻¹x1 for some λ ∈ F^×q. If NG1(hx₁i) ⊆ NG1(hy₁i), then Ay₁ =ay₁, By₁ =by₁ for somea, b∈F^×q. Hence

y₁⊗(y₁+αx₁) =g(y₁⊗(y₁+αx₁)) =aby₁⊗y₁+αaλ⁻¹(y₁⊗x₁)^σ.

Comparing the coefficients of y₁ ⊗y₁ and y₁ ⊗x₁ in the above equality we get ab = aλ⁻¹ = 1 and σ = 1. So, A=λI, B =λ⁻¹I and g = 1, as claimed. Similarly, if every non-identity element ofC_G1(x1) takesy1toy1+x1, then by multiplyingAwithλ⁻¹ and B withλ, we can assume thatλ= 1. Then for someε_a, ε_b ∈ {0,1} we have

y1⊗(y1+αx1) =g(y1⊗(y1+αx1)) =

(y1+εax1)⊗(y1+ (α+ε_b)x1) σ

. Comparing the coefficients of x₁⊗x₁, x₁⊗y₁ and y₁⊗x₁ we getε_a=ε_b = 0, σ= 1, so g= 1 follows.

Now, let q = 3. Let x₁, y₁, z₁ ∈ V₁ be a strong base for G₁. Then the stabilizer of x1⊗x1⊗ · · · ⊗x1

| {z }

tfactors

∈ V is of the form H = H1o_cSt, where y1, z1 ∈ V1 is a strong base forH₁=N_G1(x₁), so b^∗(H₁)≤2. If (m, t)6= (2,2) thenb(H)≤2 by [45, Theorem 3.6], which results inb(G)≤3. Finally, let (m, t) = (2,2). By choosing a basisx1, y1∈V1, it is easy to see that x₁⊗x₁, y₁⊗y₁, x₁⊗y₁∈V is a base for GL(V₁)o_cS₂ ≥G.

As for the order of G notice that G ≤ G1o_cS where S ≤ St is a 3-solvable group.

Thus by induction and by [80, Corollary 1.5] we have

|G| ≤ |G₁|^t|S| ≤24^−t/3|V₁|^c1t24^(t−1)/3= 24^−1/3|V|^c1.

4.9 Almost quasisimple groups

Finally, let Z ≤ N C G be such that N/Z is a non-Abelian simple group. Let N₁ = [N, N]CGand letV1be an irreducibleFpN1-submodule ofV andG1={g∈G|g(V1) = V1} be the stabilizer of V1. By using the same argument as in the last paragraph of [45, Page 29] we get thatG₁ is included in GL(V₁) and we have a chain of subgroups

N1 CG1≤GL(V1) whereG1 isp-solvable,N1 is quasisimple andV1 is irreducible as an FpN1-module.

Suppose that b(G1) ≤2 in the action of G1 on V1, that is, there exist x, y ∈V1 ≤V such that C_G1(x)∩C_G1(y) = 1. For any element g ∈ G with g(x) = x we have that N₁x = {nx|n ∈ N₁} is a g-invariant subset. As the Fp-subspace generated by N₁x is exactlyV1, we get thatg∈G1. This proves thatCG(x)∩CG(y) =CG1(x)∩CG1(y) = 1.

Thusb(G)≤2.

Hence if we manage to show that b(G1) ≤ 2 then we are finished with the proofs of both Theorems 1.6 and 1.7.

So assume that G=G₁, N =N₁, andV =V₁. By the first three paragraphs of this section, we have thatq=p. To summarize,G≤GL(V) is a group having a quasisimple irreducible normal subgroupN andZ ≤G.

We can assume thatG/Z is almost simple. For this it is sufficient to see thatN/Z is the unique minimal normal subgroup ofG/Z. For let M/Z be another minimal normal subgroup ofG/Z. By Section 4.7, we may assume thatM/Z is non-Abelian. Further-more the groupM N is a central product and so [M, N] = 1. But this is impossible since the centralizer ofN inGmust be Abelian.

Lemma 4.5. If N has a regular orbit on V thenb(G)≤2.

Proof. SinceN is normal inG, a regularN-orbit ∆ containing a given vectorvis a block of imprimitivity inside theG-orbit containingv. Hence the group CG(v)N is transitive on ∆ and N is regular on ∆. Thus for every h ∈ CG(v) the number |fix(h)| of fixed points of h on ∆ is |C_N(h)|. To prove that G has a base of size at most 2 on V, it is sufficient to see that there exists a vector w in ∆ that is not fixed by any non-trivial element ofCG(v).

First notice that if N/Z(N) is isomorphic to the non-Abelian finite simple group S then|C_G(v)| ≤ |Out(S)|< m(S) wherem(S) is the minimal index of a proper subgroup ofS. This latter inequality follows from [3, Lemma 2.7 (i)].

ButP|fix(h)|=P|C_N(h)|<|C_G(v)| ·(|N|/m(S))<|N|where the sums are over all non-identity elementsh inC_G(v). This completes the proof of the lemma.

By Lemma 4.5, in the following we may assume that N does not have a regular orbit onV. Our final theorem finishes the proofs of Theorems 1.6 and 1.7.

Theorem 4.6. Under the current assumptions Gis a p⁰-group andb(G)≤2.

Proof. By using Goodwin’s theorem [32, Theorem 1], K¨ohler and Pahlings [67, Theorem 2.2] gave a complete list of (irreducible) quasisimplep⁰-groupsN such thatN does not have a regular orbit onV. In all these exceptional cases, whenN/Zis simple,|Out(N/Z)|

is divisible by no prime larger than 3 whilepis always at least 5. SoGitself is ap⁰-group.

But thenG admits a base of size 2 on V by [45, Theorem 4.4].

5 Normalizers of primitive permutation groups

LetGbe a transitive normal subgroup of a permutation groupAof finite degreen. The factor group A/G can be considered as a certain Galois group and one would like to bound its size. One of the results of this chapter is that |A/G| < n if G is primitive unless n= 3⁴, 5⁴, 3⁸, 5⁸, or 3¹⁶. This bound is sharp when n is prime. In fact, whenG is primitive,|Out(G)|< nunlessGis a member of a given infinite sequence of primitive groups andnis different from the previously listed integers. In this chapter many other results of this flavor are established not only for permutation groups but also for linear groups.

5.1 Basic results on non-Abelian composition factors

IfG is a finite group, defineb(G) to be the product of the orders of all the non-Abelian simple composition factors of Gin a composition series for G. Two trivial observations that we shall use without comment are:

1. ifGis normal inA, thenb(A) =b(A/G)b(G); and

2. if A ≤ B, then b(A) ≤ b(B) (choose a normal series for B and intersect A with this series – Abelian quotients stay Abelian and the non-Abelian quotients can only get smaller).

The first lemma of the chapter is not used in later parts of the work, nevertheless it is worth mentioning.

Lemma 5.1. LetX₁ andX₂ be two finite groups, A≤X₁×X₂, andGA. Fori= 1,2 let πi denote the projection into Xi. (We consider πi(A) andπi(G)as subgroups of Xi.) Then

b(A/G)≤b(π₁(A)/π₁(G))b(π₂(A)/π₂(G)).

Proof. Let K denote the kernel of π₁ on A. Notice that if x ∈ π₂(K) and y ∈ π₂(G) then [x, y]∈π₂(G∩K). Henceb((π₂(K)∩π₂(G))/π₂(K∩G)) = 1. From this we get

b(K/(K∩G)) =b(π2(K)/(π2(K∩G))) =b(π2(K)/(π2(K)∩π2(G))) =

=b(π₂(K)π₂(G)/π₂(G))≤b(π₂(A)/π₂(G)).

Since b(A/G) =b(A/GK)b(GK/G) =b(π1(A)/π1(G))b(K/(K∩G)), the result follows.

The next lemma is needed for a technical result (see Theorem 5.3) for dealing with non-Abelian composition factors.

Lemma 5.2. Let J ≤Y :=X1× · · · ×Xt and assume that πi(J) =Xi for all i (where π_i is the projection onto the ith factor). Then N_Y(J)/J is solvable.

Proof. Set N = N_Y(J). Let M be the final term in the derived series of N. Let Bi = kerπ_i⁰∩J whereπ⁰_i is the projection ofY onto the direct product of all but the ith term (soB_i=J∩X_i).

Set R = B1 × · · · × Bt. Note that RJ and that R is also normal in Y since π_i(J) = X_i for all i, whence we may pass to Y /R. If we prove the result in this case, thenM R/R≤J/R, and so M ≤J. Hence we may assume from now on thatR = 1. If we prove the result in this case, thenM R/R≤J/R, whence M ≤J.

We induct on t. Ift= 1, the result is clear.

Suppose thatt= 2. SinceR = 1, we may identifyJ as a diagonal subgroup ofX1×X₂ and the normalizerN is J(Z×Z) whereZ =Z(J), whence the result.

So now assume that t > 2. By induction, we have π_i⁰(M) ≤ π⁰_i(J), whence M ≤ J(N ∩X_i). Note that [N∩X_i, X_i] = [N ∩X_i, J]≤X_i∩J = 1. Thus,M = [M, M]≤ [J(N ∩X_i), J(N∩X_i)]≤J as claimed.

Note that the proof shows that the derived length of NY(J)/J is at most t−1.

We now come to one of our major tools in studyingb(A/G).

Theorem 5.3. Assume that G C A ≤ B = XoSt = (X1 × · · · ×Xt).St = Y.St and that G acts transitively on {X₁, . . . , X_t} by conjugation. Assume that the projection of N_A(X_i) into X_i is X_i (note that N_A(X_i) ≤X_i×X_i⁰ for an obvious choice of X_i⁰). Let Ni=NG(Xi)and set Mi to be the projection of Ni into Xi. Let K be the subgroup ofA normalizing eachX_i. Then

b(A/G)≤b(A/GK)b(X₁/M₁).

Proof. We have that b(A/G) = b(A/GK)b(GK/G). So we only need to show that b(GK/G) ≤b(X₁/M₁). LetM =M₁× · · · ×M_t. LetI =J₁× · · · ×J_t whereJ_i is the projection ofJ =K∩GintoX_i.

Note first that [K, K ∩M] ≤ I (since [K, N_i] ≤ [K, G] ≤ J). In particular, (K ∩ M)/(K ∩I) is Abelian. By Lemma 5.2, (K∩I)/J is solvable. Thus, (K ∩M)/J is solvable and in particular,b((K∩M)/J) = 1. Thus,

b(GK/G) =b(K/J) =b(K/(K∩M)) =b(KM/M).

5.2 Some examples