The spread of a finite group

Abstract

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 whose socles are exceptional groups of Lie type and this is the case we treat in this paper.

Shintani descent, simple groups and spread

“…However, the asymptotic behaviour of the spread of almost simple groups is not yet known. By [10,Corollary 9], the unsettled case involves sequences of groups of Lie type defined over a field of fixed size. We settle this in our first main theorem.…”

“…This technique from algebraic groups has played an important role in the character theory of almost simple groups over several decades, beginning with the work of Shintani and Kawanaka (see [13,16,17,31,37] for example). More recently, Shintani descent has been useful in studying the almost simple groups themselves, especially in the context of spread [9,10,28,29].…”

“…It is easy to see that if s(G) > 0 then every proper quotient of G is cyclic. Recently, Burness, Guralnick and Harper [10] settled a conjecture of Breuer, Guralnick and Kantor [4], by proving that the converse is true. In fact, they proved that if every proper quotient of G is cyclic, then s(G)…”

“…A recent programme of research [3,4,9,10,24,27,28,29] has focussed on spread, a notion that captures how generating pairs are distributed across a 2-generated group (this notion is related to the generating graph and the product replacement graph, both of which have been the subject of recent research, see the discussion before [10,Corollaries 6 and 7].) Definition.…”

“…The proof involves reducing to almost simple groups. If G is simple, then s(G) 2 was proved in [4], but the almost simple groups pose further challenges and the series of papers [9,10,28,29] are dedicated to proving s(G) 2 in this case.…”