The structure of finite groups invariably generated by two elements of prime order

Abstract

A group $G$ is invariably generated by two elements $a$ and $b$ if $G=⟨a^g,b^h⟩$ for all $g,h\in G$. This paper provides a structural description of finite groups invariably generated by two elements of distinct prime orders $s$ and $t$. We illustrate the use of our main result in a case study with $s=2$ and $t=3$. In particular we prove that an almost simple group is invariably generated by an element of order $2$ and an element of order $3$ if and only if it is isomorphic to ${\mathrm{PGL}}_2(3^{2^b})$ for some $b\ge 1$.