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 with socle an exceptional group of Lie type and this is the case we treat in this paper.

Connected Components in the Invariably Generating Graph of a Finite Group

“…It is known that Γ + (G) is connected in several cases. For instance, Burness et al [11] showed that if G is a finite group such that every proper quotient of G is cyclic, then Γ(G) = Γ + (G) and Γ(G) is connected with diameter at most 2 (the special case where G is simple was proved by Guralnick and Kantor [21] and Breuer et al [3]).…”

Maximal subgroups of finite simple groups: classifications and applications

“…The only almost simple sporadic group whose maximal subgroups are not fully determined is M. The published results on maximal subgroups of M include complete classifications, except for the possibility of maximal subgroups with socle PSL 2 (8), PSL 2 (13), PSL 2 (16) and PSU 3 (4). Of these, PSL 2 (8) and PSL 2 (16) were considered in unpublished work of P. E. Holmes, and papers by Wilson on PSL 2 (8) and PSU 3 (4) are in progress. The most recent case to appear was in (99), which proves that no maximal subgroup of the Monster has socle PSU 3 (8).…”

“…In (13), Breuer, Guralnick and Kantor conjectured that the converse is also true, and in dramatic recent progress, this conjecture (and much more) has been proved by Burness, Guralnick and Harper. See their paper (16) for an excellent description of some of the history of the problem, and the key results which go into the proof. It is conjectured that there is another equivalent condition to the one above for |G| ≥ 4: the graph Γ(G) contains a Hamiltonian cycle.…”

Flexibility in generating sets of finite groups

“…3 and Burness, Guralnick, and Harper [4] completed the theorem by proving an influential conjecture of Breuer, Guralnick, and Kantor [3] regarding the case with d(G) 2.…”