ifp≥5.
The first result concerns transitive permutation groups.
Theorem 5.39. LetGCA≤Sn be transitive permutation groups of degree n. Letp be a prime. Then ap(G)|A/G| ≤cn−1.
Proof. We prove the result by induction on n.
First let A be a primitive permutation group. If A contains An, then the bound is clear. Otherwise we have |A| ≤24(n−1)/3 by [80].
Now letA be an imprimitive permutation group. Let {B1, . . . , Bt}be anA-invariant partition of the underlying set on which A acts, with 1 < t < n. Let A1 denote the action of the stabilizer of Bi in A on Bi. Then A embeds in Ai oSt and G permutes transitively the subgroups Ai. Let Gi denote the action of the stabilizer of Bi inG on Bi. By Theorem 5.3 we haveb(A/G)≤b(A/GK)b(A1/G1) whereK denotes the kernel of the action of Aon {B1, . . . , Bt}.
In order to boundap(G) we first sets=|B1|, that isn=st. We have ap(G) =ap(G/K∩G)·ap(K∩G)≤ap(GK/K)·ap(K∩G).
LetN be the kernel of the action ofK onB1. We have
ap(K∩G)·a(K/(K∩G))≤ap((K∩G)N)·a(K/(K∩G)N) =
=ap((K∩G)N/N)·ap(N)·a(K/(K∩G)N)≤ap(G1)·a(A1/G1)·ap(N).
We are now in position to boundap(G)· |A/G|=ap(G)·a(A/G)·b(A/G). We have ap(G)· |A/G| ≤ap(GK/K)· |A/GK| ·ap(K∩G)·a(K/(K∩G))·b(A1/G1)≤
≤(ap(GK/K)· |A/GK|)·(ap(G1)· |A1/G1|)·ap(N).
By induction (noting that GK/K and G1 are transitive on {B1, . . . , Bt} and B1 re-spectively), we have ap(GK/K)· |A/GK| ≤ct−1 and ap(G1)· |A1/G1| ≤ cs−1. More-over by repeated use of Proposition 5.12, we see that ap(N) ≤c(s−1)(t−1). These give ap(G)|A/G| ≤ct−1cs−1c(s−1)(t−1)=cn−1.
We next need a lemma which depends on the existence of regular orbits under certain coprime actions.
Lemma 5.40. LetAbe a primitive linear subgroup ofΓLd(pf)for a primepand integers f and d with d as small as possible. Put A0 = GLd(pf)∩A. Let J0 be the product of all normal subgroups of A contained in A0 which are nonsolvable, have orders coprime top, and are minimal with respect to being noncentral inA0. Then either pf = 7, d= 4 or 5, and b(J0) =|PSp4(3)|, or b(J0)< pf d.
Proof. The group J0 is a central product of quasisimple groups J1, . . . , Jr for some r.
Since J0 C A and A is primitive, J0 acts homogeneously on the underlying vector space. LetW be an irreducible J0-submodule. As in the proof of Theorem 5.8, we may use [62, Lemma 5.5.5, page 205 and Lemma 2.10.1, pages 47-48] to writeW in the form W1⊗ · · · ⊗Wr where for eachi theJi-module Wi (defined over a possibly larger field) is irreducible.
Assume first that r = 1. If J0 has a regular orbit on W, then the result is clear. If J0 does not have a regular orbit on W, then [100, Theorem 7.2.a] says that (J0, W) is a permutation pair in the sense of [100, Example 5.1.a] or (J0, W) is listed in the table on [100, Page 112]. In all these exceptional cases we have|J0/Z(J0)|<|W|unless W is a 4 or 5 dimensional module over the field of size 7 and J0/Z(J0)∼= PSp4(3). Moreover
|J0/Z(J0)|< pf d or|W|=pf d and one of the previous exceptional cases holds.
Assume that r > 1. For each i we have b(Ji) <|Wi| or|Wi|= 74 or 75 and b(Ji) <
|Wi|1.31. From these it follows thatb(J0)<|W| ≤pf d.
The next lemma may be viewed as a sharper version of Theorem 5.16 under the assumption thatp≥7.
Lemma 5.41. Let G be a finite group acting faithfully and completely reducibly on a finite vector space V in characteristic p. Then ap(G)≤ |V|c1/c.
Proof. By Theorem 5.16, we may assume that p≥5. By [43, Theorem 1.1], the strong base size of a p-solvable finite group S acting completely reducibly and faithfully on a vector space of sizenover a field of characteristicp≥5 is at most 2. Thus|S|< n2/(p− 1). By the proof of Theorem 5.16 we then haveap(G)≤ |V|2/(p−1)≤ |V|c1/c.
Theorem 5.39 is used in the proof of the following result which could be compared with Theorem 5.16.
Theorem 5.42. Let G C A ≤ GL(V) be linear groups acting irreducibly on a finite vector space V of size nand characteristic p. Then ap(G)|A/G| ≤24−1/3nc1.
Proof. We prove the result by induction onn.
Assume that A acts primitively on V. If ap(G) = a(G), then the result follows from part (2) of Lemma 5.32. In fact, in our argument to show Theorem 1.8 we naturally took G to be as small as possible and our calculations actually gave |A/(G∩Z(A)J)| < n apart from the eleven exceptions listed in Theorem 1.8 (when ap(G) = a(G)). Here, as usual, J denotes the central product of all normal subgroups of A contained in A0
(where A0 = GLd(pf)∩A and d is smallest with A ≤ΓLd(pf) andn =pf) subject to being noncentral. Thus we are finished if the product of the orders of the non-Abelian composition factors of J which are p0-groups is at most 24−1/3nc1−1. Let J0 be as in Lemma 5.40. By Lemma 5.40 we have b(J0) ≤ 24−1/3nc1−1 unless n ≤ 81 or pf = 7 and d = 4 or 5. It can be checked by GAP [27] that the bound holds in case n≤ 81.
5.11 Composition factors and outer automorphism groups Thus assume thatpf = 7 and d= 4 or 5 with|b(J0)|=|PSp4(3)|. Then A is a 70-group by [100, Theorem 7.2.a] and so a7(G)|A/G|=|A| ≤24−1/3nc1 by [43, Theorem 1.2].
Assume thatAacts (irreducibly and) imprimitively on V. LetV =V1⊕ · · · ⊕Vtbe a direct sum decomposition of the vector space V such that 1< t and A(and so G) acts transitively on the set {V1, . . . Vt}. Setm=|V1|. LetK be the kernel of the action of A on{V1, . . . Vt}and letA1 andG1 be the action ofNA(V1) andNG(V1) onV1respectively.
As in the proof of Theorem 5.39, we have
ap(G)· |A/G| ≤(ap(GK/K)· |A/GK|)·(ap(G1)· |A1/G1|)·ap(N),
where, in this case, N denotes the kernel of the action of K on V1. Since the groups GK/K C A/K can be viewed as transitive permutation groups acting on t points, Theorem 5.39 givesap(GK/K)|A/GK| ≤ct−1. In the proof of Theorem 5.9 it was noted thatG1must act irreducibly onV1. Thus, by the induction hypothesis,ap(G1)|A1/G1| ≤ 24−1/3mc1. Since N is subnormal in the irreducible group A, it must be completely reducible. By repeated use of Lemma 5.41 we have ap(N) ≤ mc1(t−1)/ct−1. Applying these three estimates to the displayed inequality above, we get ap(G)· |A/G| ≤ ct−1 · (24−1/3mc1)·(mc1(t−1)/ct−1) = 24−1/3nc1.
We are now in the position to complete the proof of Theorem 1.12 in the special case that Gis an affine primitive permutation group.
Theorem 5.43. Let Gbe an affine primitive permutation group of degreen, a power of a prime p. Then ap(G)|Out(G)| ≤24−1/3n1+c1.
Proof. Let H be a point stabilizer in G. We may assume that H 6= 1. Clearly H acts irreducibly and faithfully on a vector space V of size n. If H1(H, V) = {0}, then Theorem 5.42 gives the result, by Lemma 5.36. So assume that H1(H, V) 6= {0}. By Lemma 5.37,F∗(H) =L1× · · · ×LtwhereLi ∼=Lare non-Abelian simple groups viewed as subgroups of GL(Vi) where theVi are vector spaces withV =⊕ti=1Vi. For each iput
|Vi|=pdfor a primep and integerd. (See Lemma 5.37 and the proof of Theorem 5.38.) If n = q2 with q = 2e for an integer e > 1 and H = L2(q), then, by Section 5.10, a2(G)|Out(G)|< (n2logn)/2<24−1/3n1+c1. Thus, by Theorem 5.38, we may assume that |Out(G)|< n.
Assume first thatLis not p-solvable. By Lemma 5.26 and Proposition 5.12, we have the estimateap(H)≤(4d)tct−1 wherecis as in the beginning of this section, depending on p. For d = 2 and p ≥ 5, d = 3 and p ≥ 3, or d = 4 and p ≥ 3, or d ≥ 5 we have (4d)tct−1 ≤ 24−1/3pdt(c1−1) giving the desired bound for ap(H) and thus for ap(G)|Out(G)| in these cases. The only exceptions are d = 3 and p = 2, and d = 4 and p = 2. However in these cases in the previous two estimates we may replace 4d, we obtained from Lemma 5.26, by 1 and 4 respectively. Thus we may assume that Lis p-solvable (andp≥3).
Let A be the full normalizer of G in Sn. Assume that p ≥5 and that A/G is not p-solvable. By Schreier’s conjecture and the proof of Theorem 5.10 we must then havet≥ 8. By the proof of Lemma 5.41 we haveap(A)≤n3/4. Thus, by the main result of [35], Lemma 5.37 and Theorem 5.10, we have the estimateap(G)|Out(G)| ≤(n3/4)·tlogt·pd/2. Furthermore, by the proof of Theorem 5.38, we may also assume that d≥3 as well as t ≥ 8. Under these conditions it easily follows that (n3/4)·tlogt·pd/2 < 24−1/3n1+c1. Assume now thatp≥5 andA/Gisp-solvable. Then we must boundap(A)|H1(H, V)| ≤ (n3/4)·pd/2. Using Lemma 5.26 this is smaller than the desired estimate unless t≤2.
Using Lemma 5.40 we can deduce the result ift≤2 andd≥4. By the proof of Theorem 5.38 we cannot haved= 2 sinceH1(H, V)6={0} and p6= 2. Thust≤2 and d= 3. We then have by the proof of Lemma 5.41 that ap(A)|H1(H, V)| ≤ (n3/(p−1))·p which is, for n≥343, less than 24−1/3n1+c1. This forces p= 5, d= 3 and t= 1 with |L|not divisible by 5. GAP [27] shows that there is no such possibility.
We are left to consider the case whenp= 3 andLis 3-solvable, that is, a Suzuki group.
In this caseF∗(H) has a regular orbit onV by [100, Theorem 7.2.a]. We also have 1<
|H1(H, V)| ≤n1/14 by [52, Table 2] and so we may assume by Lemma 5.37 that d≥14.
By these, Proposition 5.12 and a remark after Lemma 5.25, a3(A)|H1(H, V)| < n3 <
24−1/3n1+c1 forn≥314. We may thus assume that a3(G)|Out(G)|> a3(A)|H1(H, V)|.
Then t ≥ 8 as in the previous paragraph and so a3(G)|Out(G)| < n3 ·tlogt·pd/14 <
24−1/3n1+c1 whenever n≥38·14.
Finally we finish the proof of Theorem 1.12. By Theorem 5.43 we may assume thatGis a primitive permutation group of degreenwith non-Abelian socleE which is isomorphic to a direct product of t copies of a non-Abelian simple group L. By Theorem 1.9 we know that |Out(G)|< n in this case and so it is sufficient to show ap(G) ≤24−1/3nc1 for every prime divisorpof n. This follows from the proof of Corollary 5.17.