Coprime invariable generation and minimal-exponent groups

A finite group G is coprimely invariably generated if there exists a set of generators {g₁,. .,g_u} of G with the property that the orders |g₁|,. .,|g_u| are pairwise coprime and that for all x₁,. .,x_u∈G the set {g₁^x1,. .,g_u^xu} 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,. .,π_u of the primes dividing |S|, the product of the number k_πi(S) of conjugacy classes of πi-elements satisfies. ∏_i=1u k_πi(S)≤|S|/2|OutS|.