Invariable generation with elements of coprime prime-power orders

A finite group G is coprimely-invariably generated if there exists a set of generators {g(1), ..., g(d)} of G with the property that the orders |g(1)|, ..., |g(d)| are pairwise coprime and that for all x(1), ..., x(d) is an element of G the set {g(1)(x1), ..., g(d)(xd)} generates G. In the particular case when |g(1)|, ..., |g(d)| can be chosen to be prime-powers we say that G is prime-power coprimely-invariably generated. We will discuss these properties, proving also that the second one is stronger than the first, but that in the particular case of finite soluble groups they are equivalent. (C) 2014 Elsevier Inc. All rights reserved.