Coprime invariable generation and minimal-exponent groups

A finite group G is coprimely-invariably generated} if there exists a set of generators {g(1),...,g(u)} of G with the property that the orders |g(1)|,...,|g(u)| are pairwise coprime and that for all x(1),...,x(u) is an element of G the set {g(1)(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 O-s(+) (2) which three elements. Along the way, we show that for each finite simple group S , and for each partition pi(1),...,pi(u) of the primes dividing |S| , the product of the number k(pi) (S) of conjugacy classes of pi(i) -elements satisfies