We investigate finite solvable permutation groups in which all normal subgroups are transitive or semiregular. The motivation comes from universal algebra: such groups are examples of collapsing transformation monoids.

Almost all primitive permutation groups of degree n have order at most n •, or have socle isomorphic to a direct power of some alternating group. The Mathieu groups, M11, M12, M 23 and M 24 are the four exceptions. As a corollary the sharp version of a theorem of Praeger and Saxl is established, where M 12 turns out to be the "largest" primitive group. For an application a bound on the orders of permutation groups without large alternating composition factors is given. This sharpens a lemma of Babai, Cameron, Pálfy and generalizes a theorem of Dixon.

Let G be any of the groups (P )GL(n, q), (P ) SL(n, q). Define a (simple) graph Γ = Γ (G) on the set of elements of G by connecting two vertices by an edge if and only if they generate G. Suppose that n is at least 12. Then the maximum size of a complete subgraph in Γ is equal to the chromatic number of Γ if n ≡ 2 (mod 4), or if n ≡ 2 (mod 4), q is odd and G = (P ) SL(n, q). This work was motivated by a question of Blackburn.

Let G be a group that is a set-theoretic union of finitely many proper subgroups. Cohn defined (G) to be the least integer m such that G is the union of m proper subgroups. Tomkinson showed that (G) can never be 7, and that it is always of the form q + 1 (q a prime power) for solvable groups G. In this paper we give exact or asymptotic formulas for (S n ). In particular, we show that (S n ) 2 n−1 , while for alternating groups we find (A n ) 2 n−2 unless n = 7 or 9. An application of this result is also given.

Abstract. For a finite group G let Γ(G) denote the graph defined on the nonidentity elements of G in such a way that two distinct vertices are connected by an edge if and only if they generate G. In this paper it is shown that the graph Γ(G) contains a Hamiltonian cycle for many finite groups G.