Let G be a finite simple group. By a theorem of Guralnick and Kantor, G contains a conjugacy class C such that for each non-identity element x ∈ G, there exists y ∈ C with G = x, y . Building on this deep result, we introduce a new invariant γu(G), which we call the uniform domination number of G. This is the minimal size of a subset S of conjugate elements such that for each 1 = x ∈ G, there exists s ∈ S with G = x, s . (This invariant is closely related to the total domination number of the generating graph of G, which explains our choice of terminology.) By the result of Guralnick and Kantor, we have γu(G) |C| for some conjugacy class C of G, and the aim of this paper is to determine close to best possible bounds on γu(G) for each family of simple groups. For example, we will prove that there are infinitely many non-abelian simple groups G with γu(G) = 2. To do this, we develop a probabilistic approach, based on fixed point ratio estimates. We also establish a connection to the theory of bases for permutation groups, which allows us to apply recent results on base sizes for primitive actions of simple groups.

A group G is said to be 3 2 -generated if every nontrivial element belongs to a generating pair. It is easy to see that if G has this property then every proper quotient of G is cyclic. In this paper we prove that the converse is true for finite groups, which settles a conjecture of Breuer, Guralnick and Kantor from 2008. In fact, we prove a much stronger result, which solves a problem posed by Brenner and Wiegold in 1975. Namely, if G is a finite group and every proper quotient of G is cyclic, then for any pair of nontrivial elements x1, x2 ∈ G, there exists y ∈ G such that G = x1, y = x2, y . In other words, s(G) 2, where s(G) is the spread of G. Moreover, if u(G) denotes the more restrictive uniform spread of G, then we can completely characterise the finite groups G with u(G) = 0 and u(G) = 1. To prove these results, we first establish a reduction to almost simple groups. For simple groups, the result was proved by Guralnick and Kantor in 2000 using probabilistic methods and since then the almost simple groups have been the subject of several papers. By combining our reduction theorem and this earlier work, it remains to handle the groups with socle an exceptional group of Lie type and this is the case we treat in this paper.

The distinguishing number of G Sym(Ω) is the smallest size of a partition of Ω such that only the identity of G fixes all the parts of the partition. Extending earlier results of Cameron, Neumann, Saxl and Seress on the distinguishing number of finite primitive groups, we show that all imprimitive quasiprimitive groups have distinguishing number two, and all non-quasiprimitive semiprimitive groups have distinguishing number two, except for GL(2, 3) acting on the eight non-zero vectors of F 2 3 , which has distinguishing number three.

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