In this section the second half of Theorem 1.8 and Theorem 1.10 are proved.
Let Gbe a primitive permutation group of degree nand let A be the full normalizer of Gin Sn. Assume that A is not an (affine) subgroup of AΓL1(q) for a prime power q equal ton. We will show that forn≥214000we have |A/G|< n1/2logn.
Assume first thatAis an affine primitive permutation group. Then so isG. In fact we may change notation and assume thatAandGare linear groups acting irreducibly on a finite vector spaceV of size nwith GCA. First assume that A acts primitively onV. Let us use the notation of Theorem 5.30. By assumption, we haved≥2. If d= 2 then, by the structure ofA, we have that |A/G|is at most (3/2)(√
n−1) log3n < n1/2logn ifA is solvable and|A/G|< n1/2lognifA is nonsolvable (using Dickson’s theorem). If
5.12 Asymptotics
d≥3, then, by Theorem 5.30,
|A/G|< n1/3·(logn)·d2 logd+3 ≤n1/3(logn)2 log logn+4 < n1/2 forn≥28192.
Assume now thatA acts imprimitively on V and that it preserves a direct sum de-composition V1 ⊕ · · · ⊕Vt of V where t > 1 is as large as possible. Let K denote the kernel of the (transitive) action of A on {V1, . . . , Vt}. As in the proof of Theorem 5.28, let A1 be the action ofNA(V1) onV1. The groupA1 acts primitively and irreducibly on V1. By Theorem 5.27, we have a(A1) < mlogm form=|V1|>316. In the notation of Lemma 5.20 for t≥2729 we havea(K/J) ≤(a(X1))t/3c1, by use of Lemma 5.23, where c1 is as in Theorem 1.12. From this and by the proof of Theorem 5.28 together with part (2) of Lemma 5.32 and Theorem 1.14, for t≥2729 we get
|A/G| ≤ |A/GK| ·(a(A1)b(A1/G1))t/3c1 ≤16t/
√logt·n1/3 < n1/2.
Otherwise, if t is bounded (is at most 2729) but t 6= 2, 4, then, again by the proof of Theorem 5.28 and by Lemma 5.20 and Theorems 1.14 and 5.9, we have, forn≥28192,
|A/G| ≤ |A/GK| ·b(A1/G1)·(a(A1))3t/8 ≤16t/
√logt·(logm)2 log logm·n7/16< n1/2.
Ift= 2 and m≥22048, then|A/G| ≤b(A1/G1)a(A1)< mlogm < n1/2logn.
Let t = 4. Then |A/G| ≤ 6·b(A1/G1)·(a(A1))2. Using the notation d (for A1) as in Theorem 5.30, by the argument above, we see that ford≥2 and m≥22048 we have b(A1/G1)·a(A1) < 16·m1/2logm. This gives |A/G|< n1/2logn for n ≥ 28192 under the assumption thatd≥2. Thus assume thatd= 1. Here we use the observation made in Lemma 5.21. Write a(A1) =|A1|in the form 2`r where r is odd and `is an integer.
Then we have|A/G| ≤6·22`·r. From this the result follows if |A1|<6m. Otherwise, by Zsigmondy’s theorem, 2` < m. Now|A1|< mlogqm where q is the size of the field over whichA1andAare defined. From this 22`·r < m2logqmgiving|A/G|< n1/2logn unless q = 2. If q = 2, then 2` ≤logm, and so |A/G| <6·m(logm)2 < n1/2logn for m≥22048.
Assume thatAis a primitive permutation group which is not of affine type. In this case we use the notation, assumptions and the argument in the last two paragraphs of Section 5.7. However, we use Theorem 1.14. If A has two minimal normal subgroups, then we have |A/G| = a(A/G)b(A/G) < n1/3 ·4s/
√logs ·(logn)2 log logn < n1/2 for n ≥ 28192, if s 6= 1, and also when s = 1. Finally assume that A has a unique minimal normal subgroup. We first claim that we may assume that t ≥ 512. If |Out(L)| ≤ √
m and t≤512, then|A/G| ≤ n1/4·16t/
√logt< n1/2. If|Out(L)|is larger than √
m, then L is one of the exceptions in Lemma 5.25 and so t ≥512 by use of n ≥214000. If t ≥512, then, for n≥214000, we have |A/G| ≤ |Out(L)|1/3t·4t/
√logt·tlogt< n1/2logn.
This proves the second half of Theorem 1.8.
So far we showed Theorem 1.10 in case |Out(G)| = |A/G|. In fact by the same calculation as in the previous paragraph we established Theorem 1.10 in case G is not of affine type. Thus assume that Gis of affine type and H1(H, V) 6= 0 where V is the minimal normal subgroup of G and H 6= 1 is a point stabilizer in G. Let us use the notation and assumptions of the proof of Theorem 5.38.
Let us first assume that t > 1 and |Out(G)| ≤(q−1)q[d/2]|Out(L)|t/2bt with d≥ 2 wherebt≤4t/
√logt·tlogt, by Theorem 1.14 and|Out(L)| ≤4(logn)/t, by Lemma 5.26.
Here n = qdt. Thus |Out(G)| < q[d/2]+1 ·(4(logn)/t)t/2 ·4t/
√logt·tlogt. If d ≥ 3 or t≥3, then this is less than n1/2logn for n≥214000. Finally, if d= 2 andt = 2, then
|Out(G)|< n1/2logn since|Out(L)| ≤lognby use of Dickson’s theorem on subgroups Lof GL2(q).
Finally assume that in the proof of Theorem 5.38 we have t = 1, that is, H is an almost simple group with socle L. Then |Out(G)| ≤(q−1)|H1(L, W)||Out(L)|where W is a nontrivial irreducibleL-module of size dividingnand defined over a field of size q. By Lemma 5.26,|Out(L)| ≤4 logn.
By the main result of [35], we have|Out(G)| ≤4(q−1)|W|1/2logn. Assume first that
|W|< n. If dimW ≥3, then |Out(G)| ≤4·n5/12logn, which is less than n1/2lognfor n≥ 214000. If dimW = 2, then |Out(G)|< n1/2|Out(L)| which is less than n1/2logn by using Dickson’s theorem once again. Thus assume that|W|=n=qd. Furthermore, as observed in the proof of Theorem 5.38, we may assume that d≥3 (otherwise Gis a member of the infinite sequence of examples in Theorem 1.9). Then, by [35], it is easy to see that |Out(G)| < 2·n3/4, at least for n ≥ 214000. (In this previous bound the factor 2 comes from the fact that the full normalizer in GL4(q) of A6 acting on the fully deleted permutation module of dimension 4 in characteristic 3 over the field of sizeq has size (q−1)|S6|since the dimension of the fixed point space of a 3-cycle in A6 is different from the dimension of the fixed point space of an element in A6 which is the product of two 3-cycles.) This proves the first statement of Theorem 1.10.
If L is an alternating group of degree at least 7, a sporadic simple group or the Tits group, then dim(H1(L, W)) ≤ (1/4) dim(W) by [36, Corollary 3] and [35]. Thus in these cases we have |Out(G)| < 2 ·n1/2 ≤ n1/2logn. If L is a simple group of exceptional type, then dim(H1(L, W)) ≤ (1/3) dim(W) by [52], thus if d ≥ 7 then
|Out(G)|<4·n3/7logn < n1/2lognforn≥214000. Otherwised= 6 andL= G2(r) with reven or 4≤d≤6 andLis a Suzuki group (by [62, Table 5.3.A] and [62, Table 5.4.C]).
However in these cases dim(H1(L, W))≤1, by [52], and so|Out(G)|< n1/2logn. We may now assume thatL is a classical simple group.
Since we are assuming that the dimension of the natural module for L is at least 7, we see from [62, Table 5.4.C] and [62, Table 5.3.A] that d ≥ 7. By [52] we also have dim(H1(L, W)) ≤(1/3) dim(W). Thus |Out(G)|< 4·n3/7logn < n1/2logn for n≥214000.
This completes the proof of Theorem 1.10.
6 Fixed point spaces
In this last chapter of the thesis we consider fixed point spaces of elements of linear groups. Let Gbe a finite group, F a field, andV a finite dimensional F G-module such that Ghas no trivial composition factor on V. Then the arithmetic average dimension of the fixed point spaces of elements of G on V is at most (1/p) dimV where p is the smallest prime divisor of the order ofG. This answers and generalizes a 1966 conjecture of Neumann [86] which also appeared in a paper of Neumann and Vaughan-Lee [87]
and also as a problem in The Kourovka Notebook posted by Vaughan-Lee. Our result also generalizes a recent theorem of Isaacs, Keller, Meierfrankenfeld, and Moret´o [57].
We also classify precisely when equality can occur. Various applications are given. For example, another conjecture of Neumann and Vaughan-Lee is proven and some results of Segal and Shalev [103] are improved and/or generalized concerning BFC groups.
6.1 The proofs
Our first lemma sharpens and generalizes [87, Theorem 6.1].
Lemma 6.1. Let Gbe a finite group,F a field, and V a finite dimensionalF G-module.
Let N be an elementary Abelian normal subgroup of G such that CV(N) = 0. Then avgdim(N g, V) ≤ (1/p) dimV for every g ∈ G where p is the smallest prime factor of the order of G.
Proof. We may assume thatF is algebraically closed. Let us consider a counterexample with |G| and dimV minimal. It clearly suffices to assume that G = hg, Ni. We may assume that V is irreducible (since if we have the inequality on each composition factor of V we have it on V). Finally, we may assume thatN acts faithfully on V. If N does not act homogeneously on V, then g transitively permutes the components in an orbit of size t≥p and so every element in N g has a fixed point space of dimension at most (1/t) dimV ≤ (1/p) dimV. So we may assume that the elementary Abelian group N acts homogeneously on V. This means that it acts as scalars on V. Thus N ≤ Z(G) andG/Z(G) is cyclic. It follows thatGis Abelian and so dimV = 1. At most 1 element in the coset N g is the identity and so avgdim(N g, V) ≤ (1/|N|) dimV ≤(1/p) dimV. The result follows.
We first need a result about generation of finite groups. This is an easy consequence of the proof of the main results of [8].
Theorem 6.2. LetGbe a finite group with a minimal normal subgroupN =L1×. . .×Lt for some positive integer twith Li ∼=L for alli with1≤i≤tfor a non-Abelian simple group L. Assume that G/N = hxNi for some x ∈ G. Then there exists an element s∈L1≤N such that |{g∈N x:G=hg, si}|>(1/2)|N|.
Proof. First suppose that t= 1. This is an immediate consequence of [8, Theorem 1.4]
unlessGis one of Sp2n(2), n >2,S2m+1 orL= Ω+8(2) orA6.
If G= Sp2n(2), n > 2, then the result follows by [8, Proposition 5.8]. IfG =S2m+1, then apply [8, Proposition 6.8].
Suppose that L=A6. Note that the proper overgroups of sof order 5 in A6 are two subgroups isomorphic to A5 (of different conjugacy classes) and the normalizer of the subgroup generated bys. The result follows trivially from this observation.
Finally considerL= Ω+8(2). We takesto be an element of order 15. It follows by the discussion in [8, Section 4.1] that givenG, there is an element of order 15 satisfying the result (although it is possible that the choice ofsdepends on which Goccurs).
Now assume thatt >1. Writex= (u1, . . . , ut)σ whereσ just cyclically permutes the coordinates ofN (sendingLi toLi+1 fori < t) andui ∈Aut(Li). By conjugating by an element of the group Aut(L1)×. . .×Aut(Lt) we may assume that u2 =. . . = ut = 1 (we do not need to do this but it just makes the computations easier).
Letf :N x→Aut(L1) be the map sendingwxto the projection of (wx)t in Aut(L1).
Write w = (w1, . . . , wt) with wi ∈ Li. Then f(wx) = wtwt−1. . . w1u1 is in L1u1. Moreover, we see that every fiber of f has the same size. By the case t= 1, we know that the probability thathf(wx), si=hL1, u1i is greater than 1/2.
We claim that ifL1 ≤ hf(wx), si, thenG=hwx, si. The claim then implies the result.
So assume thatL1 ≤ h(f(wx), siand setH=hwx, si. LetM ≤N be the normal closure of s in J := h(wx)t, si. This projects onto L1 by assumption, but is also contained in L1, whence M =L1. So L1 ≤H. Since any element ofN x acts transitively on the Li, it follows thatN ≤H and so G=H.
The next result we need is Scott’s Lemma [101].
Lemma 6.3 (Scott’s Lemma). Let G be a subgroup of GL((V)) with V a finite dimen-sional vector space. Suppose that G=hg1, . . . , gri with g1· · ·gr= 1. Then
r
X
i=1
dim[gi, V]≥dimV + dim[G, V]−dimCV(G).
We will apply this in the caser = 3. Noting that dimV = dim[x, V] + dimCV(x) for any x, we can restate this as:
6.1 The proofs
3
X
i=1
dimCV(gi)≤dimV + dimCV(G) + dimV /[G, V].
Theorem 6.4. Let G be a finite group. Assume that G has a normal subgroup E that is a central product of quasisimple groups. Let V be a finite dimensional F G-module for some field F such that E has no trivial composition factor on V. If g ∈ G, then avgdim(gE, V)≤(1/2) dimV.
Proof. Let us consider a counterexample with |G| and dimV minimal. There is no loss of generality in assuming that F is algebraically closed, G = hE, gi, and then assuming thatV is an irreducible (hence absolutely irreducible) and faithfulF G-module.
IfZ(E)6= 1, the result follows by Lemma 6.1 (by takingN =Z(E) and noting thatZ(E) is completely reducible on V with CV(Z(E)) = 0 (since V is a faithful F G-module)).
So we may assume that E is a direct product of non-Abelian simple groups. If V is not a homogeneousF E-module, theng transitively permutes the homogeneous components and so any element ingE has fixed point space of dimension at most (1/2) dimV. So we may assume thatV is a homogeneousF E-module. ThusE =L1×. . .×Lm with theLi’s non-Abelian simple groups. So V is a direct sum of sayt copies of V1⊗. . .⊗Vm where Vi is an irreducible nontrivialF Li-module. (Since G/E is cyclic andV is irreducible, it follows thatt= 1 (by Clifford theory) but we will not use this fact.) We may replaceEby a minimal normal subgroup ofGcontained inE (the hypothesis on the minimal normal subgroup will hold by Clifford’s theorem) and so assume that g transitively permutes the isomorphic subgroups L1, . . . , Lm.
Let s∈ L1 ≤ E be chosen so that Y := {y ∈ gE : hy, si = G} has size larger than (1/2)|E|. Such an element exists by Theorem 6.2. Setc= dimCV(s). Ify ∈Y then, by Lemma 6.3 (applied to the triple (y, s,(ys)−1)), we have
c+ dimCV(y) + dimCV(ys)≤dimV.
For anyy ∈Y0 :=gE\Y, we at least have
dimCV(y) + dimCV(ys)≤dimV +c.
Thus,
2 X
y∈gE
dimCV(y) = X
y∈gE
dimCV(y) + dimCV(ys) is at most
|Y|(dimV −c) +|Y0|(dimV +c)<|E|dimV.
This gives avgdim(gE)≤(1/2) dimV as required.
We now prove Theorem 1.16. As usual, we may assume thatF is algebraically closed, V is an irreducibleF G-module, andN acts faithfully onV. LetAbe a minimal normal
subgroup ofG contained in N. Since V is a faithful completely reducible F N-module, A has no trivial composition factor on V. Now apply Lemma 6.1 and Theorem 6.4 to conclude that avgdim(Ag, V)≤(1/p) dimV wherepis the smallest prime divisor of|G|.
SinceN g is the union of cosets of A, the result follows.
We next consider fields of characteristic 0.
Lemma 6.5. Let G be a finite group, C the field of complex numbers, and V a finite dimensional CG-module. For an element g∈G and a root of unity a∈C let ag denote the multiplicity of a as an eigenvalue of g. Then P
g∈Gag =P
g∈Gbg as long as a and b have the same order inC∗.
Proof. Let a and b be roots of unity of the same order. Let m be the exponent of G withµa primitive m-th root of unity. Let σ be an element of the automorphism group of the field Q(µ) withσ(a) = b. Letebe a positive integer such that σ(µ) =µe. Then
The M¨obius function µ(n) of a positive integer n is 0 if n is not square free and is (−1)m if n is square free and the number of (distinct) prime divisors of n is m. For a positive integern let s(n) be the sum of primitiventh roots of unity (in C). We recall the following well known result.
Lemma 6.6. For a positive integer n we have s(n) =µ(n).
Proposition 6.7. Let G be a finite group, let F be a field such that |G| is invertible in F, let V be a finite dimensional F G-module with no trivial F G-composition factor, and letp be the smallest prime divisor of the order ofG/CG(V). Then avgdim(G, V)≤
Lettingϕ(n) denote the Euler function ofn, we may write the previous equation as 0 = with equality if and only if the exponent ofG isp.
6.1 The proofs Now we prove Theorem 1.17. By Proposition 6.7, we know that equality always occurs when G/CG(V) is a group of exponent p. Hence, it remains to show that whenever avgdim(G, V) = (1/p) dimV, thenG/CG(V) is a group of exponent p.
Choose a minimal counterexample to this latter statement with respect to |G| and dimV. As before, we may assume that CG(V) = 1. By Proposition 6.7, we may also assume thatr := char(F) divides the order of G.
We claim thatV is an irreducibleF G-module. For suppose not andW is a non-trivial proper submodule of V. By the minimality of dimV and by the fact that
avgdim(G, V)≤avgdim(G, W) + avgdim(G, V /W)≤(1/p) dimW + (1/p) dimV /W, we have that G/CG(W) and G/CG(V /W) are groups of exponent p. Let N be the normal subgroup of G which acts trivially on both W and V /W. Note that N is an r-group. So G = P N where P is a Sylow p-subgroup of G of exponent p. Since G is a counterexample to the above statement, N 6= 1. For any element g ∈ P we have avgdim(gN, V) ≤dimCV(g). (This can be seen by observing that some power of an arbitrary element gn is conjugate to g. Moreover, avgdim(N, V) ≤ (1/r) dimV <
(1/p) dimV. Thus,
avgdim(G, V) =|P|−1X
g∈P
avgdim(gN, V)<avgdim(P, N) = (1/p) dimV, a contradiction.
So we may assume thatV is an irreducible F G-module. Let M be a minimal normal subgroup ofG. By Theorem 1.16, we have avgdim(M g, V)≤(1/p) dimV for each coset M g of M in G, so avgdim(M g, V) = (1/p) dimV must hold for each coset M g of M in G. In particular, by the minimality of G, the group M is an elementary Abelian p-group. Since Gis not a p-group, we can choose g∈Gof prime orders > p such that G=hg, Mi(by the minimality ofG). (The moduleV remains an irreducibleF G-module (by the minimality of dimV) andCG(V) = 1 continues to hold since bothM and gacts faithfully on V.) If M is central in G, then G is Abelian and dimV = 1. In this case avgdim(G, V) = (1/|G|) dimV <(1/p) dimV, a contradiction. IfM is not central, then g permutes the eigenspaces of M in an orbit of size s > p (for some divisort of s) and so avgdim(M g, V) ≤ (1/t) dimV < (1/p) dimV, which is again a contradiction. This proves Theorem 1.17.
Let us next prove the first statement of Corollary 1.19. By making the assumptions of the proof of [57, Corollary D], it is sufficient to show that the number of g∈G such that dimCV(g)≤(1/2) dimV is at least
2|G|
1 + logp|G|p ≤ 2|G|
2 + dimV. But this is clear by Theorem 1.16 noting that dimV is even.
Let us prove the second statement of Corollary 1.19. Use the notations and assump-tions of the last part of the proof of [57, Corollary D]. LetH be a Hall p0-subgroup of G. SinceV is a completely reducibleG-module with CV(G) = 0, the vector space V is also a completely reducibleH-module with CV(H) = 0. Hence applying Corollary 1.18 to the H-module V we get that there exists g ∈H with dimCV(g)< (1/2) dimV. So the last displayed inequality of the proof of [57, Corollary D] becomes
|clG(g)|p
p ≥χ(1)1/3
since dimV is even. From this we get that p3χ(1)≤ |clG(g)|p3. In the next two paragraphs we prove Theorem 1.20.
Note that Y centralizes M and so there is no loss in working inG/Y and assuming thatX=M is a minimal normal subgroup ofG. SetH=hM, gi and so assume thatg acts transitively on the direct factors ofM.
We compute the arithmetic mean of the positive integers |CM(x)| for x ∈ gM. All elements in a givenM-conjugacy class ingM have the same centralizer size. Ifh∈gM, then the M-conjugacy class of h has |M : CM(h)| elements. Thus, we see that the arithmetic mean is precisely the number ofM-conjugacy classes ingM. By [25, Lemma 2.1], this is at mostk(M), the number of conjugacy classes in M. By [25, Proposition 5.3], this is at most|M|.41. Since the geometric mean is bounded above by the arithmetic mean, the result follows.
We end this chapter by proving Theorem 1.21.
Let us fix a chief series for a finite group G. Let N be the set of non-central chief factors of this series. Let p be the smallest prime factor of the order of G/F(G). If N ∈ N is Abelian then, by Theorem 1.16 (noting that F(G) acts trivially on N), we have geom(G, N) ≤ |N|1/p. If N ∈ N is non-Abelian then, by Theorem 1.20 and the Feit-Thompson Odd Order Theorem [22], we again have geom(G, N) ≤ |N|1/p. Notice also that for any g ∈G we have the inequality |CG(g)| ≤ ccf(G)Q
Bibliography
[1] M. Aschbacher, Finite group theory. Second edition. Cambridge Studies in Ad-vanced Mathematics, 10. Cambridge University Press, Cambridge, 2000.
[2] M. Aschbacher and R. M. Guralnick, Some applications of the first cohomology group.J. Algebra 90(1984), 446–460.
[3] M. Aschbacher and R. M. Guralnick, Abelian quotients of primitive groups.Proc.
Amer. Math. Soc. 107 (1989), 89–95.
[4] B. Baumeister, A. Mar´oti, H. P. Tong-Viet, Finite groups have more conjugacy classes. Forum Math.29(2017), no. 2, 259–275.
[5] R. Brauer, On groups whose order contains a prime number to the first power. I.
Amer. J. Math.64, (1942), 401–420.
[6] R. Brauer, Representations of finite groups. 1963 Lectures on Modern Mathemat-ics, Vol. I pp. 133–175 Wiley, New York.
[7] R. Brauer and W. Feit, On the number of irreducible characters of finite groups in a given block, Proc. Natl. Acad. Sci. USA 45(1959) 361–365.
[8] T. Breuer, R. M. Guralnick, W. M. Kantor, Probabilistic generation of finite simple groups, II. J. Algebra Vol. 320.2, (2008), 443–494.
[9] M. Cartwright, The order of the derived group of a BFC group.J. London Math.
Soc. 30(1984), 227–243.
[10] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, Wilson, R. A. Atlas of finite groups.Oxford University Press, Eynsham, (1985).
[11] J. P. Cossey, Z. Halasi, A. Mar´oti, H. N. Nguyen, On a conjecture of Gluck.Math.
Z.279 (2015), no. 3-4, 1067–1080.
[12] J. D. Dixon, The Fitting subgroup of a linear solvable group. J. Austral. Math.
Soc. 7 (1967), 417–424.
[13] J. D. Dixon and B. Mortimer, Permutation groups, Graduate Texts in Math. 163, Springer-Verlag, New York, 1996.
[14] S. Dolfi, Orbits of permutation groups on the power set. Arch. Math. 75 (2000), 321–327.
[15] S. Dolfi, Intersections of odd order Hall subgroups. Bull. London Math. Soc. 37 (2005) 61–66.
[16] S. Dolfi, Large orbits in coprime actions of solvable groups. Trans. Amer. Math.
Soc.360 (2008), 135–152.
[17] S. Dolfi and E. Jabara, Large character degrees of solvable groups with Abelian Sylow 2-subgroups.J. Algebra 313(2007), 687–694.
[18] Y. A. Drozd and V. V. Kirichenko, Finite-dimensional algebras. Springer-Verlag, Berlin, 1994.
[19] P. Erd˝os, On an elementary proof of some asymptotic formulas in the theory of partitions.Ann. of Math.(2) 43, (1942). 437–450.
[20] P. Erd˝os and P. Tur´an, On some problems of a statistical group-theory. IV. Acta Math. Acad. Sci. Hungar 19(1968), 413–435.
[21] J. A. Ernest, Central intertwining numbers for representations of finite groups.
Trans. Amer. Math. Soc.99(1961), 499–508.
[22] W. Feit and J. G. Thompson, Solvability of groups of odd order.Pacific J. Math.
13(1963), 775-1029.
[23] G. A. Fern´andez-Alcober and M. Morigi, Outer commutator words are uniformly concise.J. Lond. Math. Soc.(2)82(2010), no. 3, 581–595.
[24] D. A. Foulser, Solvable primitive permutation groups of low rank. Trans. Amer.
Math. Soc.143 (1969), 1–54.
[25] J. Fulman and R. M. Guralnick, Bounds on the number and sizes of conjugacy classes in finite Chevalley groups with applications to derangements.Trans. Amer.
Math. Soc.364 (2012), no. 6, 3023–3070.
[26] P. X. Gallagher, The number of conjugacy classes in a finite group.Math. Z. 118 (1970) 175–179.
[27] The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.4; 2005,(http://www.gap-system.org).
[28] M. Garonzi and A. Mar´oti, On the number of conjugacy classes of a permutation group.J. Combin. Theory Ser. A 133 (2015), 251–260.
[29] D. Gluck, Trivial set-stabilizers in finite permutation groups.Canad. J. Math. 35 (1983), no. 1, 59–67.
[30] D. Gluck and K. Magaard, Base sizes and regular orbits for coprime affine permu-tation groups.J. London Math. Soc. (2)58(1998), 603–618.
[31] D. Gluck, K. Magaard, U. Riese, P. Schmid, The solution of the k(GV)-problem.
J. Algebra 279(2004), no. 2, 694–719.
[32] D. P. M. Goodwin, Regular orbits of linear groups with an application to the k(GV)-problem, I,II. J. Algebra 227 (2000), 395–432 and 433–473.
[33] R. M. Guralnick, Generation of simple groups.J. Algebra 103(1986), 381–401.
Bibliography [34] R. M. Guralnick, Cyclic quotients of transitive groups. J. Algebra 234 (2000),
507–532.
[35] R. M. Guralnick and C. Hoffman, The first cohomology group and generation of simple groups, in Proc. Conf. Groups and Geometries (Siena, 1996), Trends in Mathematics, Birkh¨auser, Basel 1998, pp. 81–89.
[36] R. M. Guralnick and W. Kimmerle, On the cohomology of alternating and sym-metric groups and decomposition of relation modules. J. Pure Appl. Algebra 69 (1990), no. 2, 135–140.
[37] R. M. Guralnick and G. Malle, Products of conjugacy classes and fixed point spaces. J. Amer. Math. Soc.25(2012), no. 1, 77–121.
[38] R. M. Guralnick and A. Mar´oti, On the non-coprime k(GV)-problem.J. Algebra 385 (2013), 80–101.
[39] R. M. Guralnick and A. Mar´oti, Average dimension of fixed point spaces with
[39] R. M. Guralnick and A. Mar´oti, Average dimension of fixed point spaces with