Let $G$ be a finite almost simple group. It is well known that $G$ can be generated by three elements, and in previous work we showed that 6 generators suffice for all maximal subgroups of $G$. In this paper, we consider subgroups at the next level of the subgroup lattice—the so-called second maximal subgroups. We prove that with the possible exception of some families of rank 1 groups of Lie type, the number of generators of every second maximal subgroup of $G$ is bounded by an absolute constant. We also show that such a bound holds without any exceptions if and only if there are only finitely many primes $r$ for which there is a prime power $q$ such that $(q^{r}-1)/(q-1)$ is prime. The latter statement is a formidable open problem in Number Theory. Applications to random generation and polynomial growth are also given.

Let $G=G(q)$ be a finite almost simple exceptional group of Lie type over the field of $q$ elements, where $q=p^a$ and $p$ is prime. The main result of the paper determines all maximal subgroups $M$ of $G(q)$ such that $M$ is an almost simple group which is also of Lie type in characteristic $p$, under the condition that ${\rm rank}(M) > {1\over 2}{\rm rank}(G)$. The conclusion is that either $M$ is a subgroup of maximal rank, or it is of the same type as $G$ over a subfield of $\F_q$, or $(G,M)$ is one of $(E_6^\e(q),F_4(q))$, $(E_6^\e(q),C_4(q))$, $(E_7(q),\,^3\!D_4(q))$. This completes work of the first author with Saxl and Testerman, in which the same conclusion was obtained under some extra assumptions.

For a finite group $H$, let $Irr(H)$ denote the set of irreducible characters of $H$, and define the ‘zeta function’ $\zeta^H(t) = \sum_{\chi \in Irr(H)} \chi(1)^{-t}$ for real $t > 0$. We study the asymptotic behaviour of $\zeta^H(t)$ for finite simple groups $H$ of Lie type, and also of a corresponding zeta function defined in terms of conjugacy classes. Applications are given to the study of random walks on simple groups, and on base sizes of primitive permutation groups.

Dedicated to the memory of Helmut Wielandt

In this paper a classification is obtained of all pairs G, H of finite primitive permutation groups such that G < H, the socles of G and H are different, and G and H share a common non-trivial orbital. An application is given to the study of automorphism groups of edge-transitive graphs.

2000 Mathematical Subject Classification: 20B15, 20B25.

A proof is given that for primes r, s, not both 2, and for finite simple classical groups G of sufficiently large rank, the probability that two randomly chosen elements in G of orders r and s generate G tends to 1 as |G| → ∞.

Let G be a simple algebraic group over an algebraically closed field K of characteristic p. If Σ is the root system of G and Uα is the root subgroup of G corresponding to a long root α∈Σ, then 〈Uα, U−α〉 is an image of SL2 and any G-conjugate of this subgroup is called a fundamental subgroup of G. In [LS1], the closed connected semisimple subgroups of G generated by long root elements were determined. Of course, long root elements are unipotent elements of fundamental subgroups. In this paper we consider subgroups of G which are generated by semisimple elements lying in fundamental subgroups.

Our first three results (Theorems 1–3) concern semisimple connected subgroups of G which contain a maximal torus of a fundamental subgroup; we call such a torus a fundamental torus. Notice that for classical groups, the elements in fundamental tori have fixed spaces of small codimension in the natural module; indeed, for the groups SL(V) and Sp(V), the elements of fundamental tori are precisely those semisimple elements with fixed space of codimension 2, while for SO(V), the codimension is 4.

As consequences of these results, we obtain information on subgroups (finite or infinite) of classical groups which are generated by conjugates of a single element of a fundamental torus of order at least 5 (see Theorems 4, 5 and Corollary 6); for example, Theorem 5 determines those finite irreducible subgroups containing such an element which are quasisimple and of Lie type in characteristic p.

We now state our results in detail. Recall from [LS1] that a subsystem subgroup of G is a connected semisimple subgroup which is invariant under a maximal torus of G.

Let G be a finite group acting faithfully and transitively on a set Ω of size m, and let E = {x1, …, xk} be a generating set for G with x1x2…xk = 1. If x∈G has cycles of length r1, …, rl in its action on Ω, define ind (x) = [sum ]l1(ri−1). Then the genusg = g(G, Ω, E) is defined by

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