Coprime invariable generation and minimal-exponent groups

Abstract

A finite group G is coprimely invariably generated if there exists a set of generators {g1,…,gu} of G with the property that the orders |g1|,…,|gu| are pairwise coprime and that for all x1,…,xu∈G the set {g1x1,…,guxu} generates G.We show that if G is coprimely invariably generated, then G can be generated with three elements, or two if G is soluble, and that G has zero presentation rank. As a corollary, we show that if G is any finite group such that no proper subgroup has the same exponent as G, then G has zero presentation rank. Furthermore, we show that every finite simple group is coprimely invariably generated by two elements, except for O8+(2) which requires three elements.Along the way, we show that for each finite simple group S, and for each partition π1,…,πu of the primes dividing |S|, the product of the number kπi(S) of conjugacy classes of πi-elements satisfies∏i=1ukπi(S)≤|S|2|OutS|.