Invariable generation with elements of coprime prime-power order

Abstract

A finite group $G$ is coprimely-invariably generated if there exists a set of generators $\{g_1, \ldots, g_d\}$ of $G$ with the property that the orders $|g_1|, \ldots, |g_d|$ are pairwise coprime and that for all $x_1, \ldots, x_d \in G$ the set $\{g_1^{x_1}, \ldots, g_d^{x_d}\}$ generates $G$. In the particular case when $|g_1|, \ldots, |g_d|$ 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.