Published online by Cambridge University Press: 20 November 2015
Motivated by a problem of characterising a family of Cayley graphs, we study a class of finite groups $G$ which behave similarly to elementary abelian
$p$-groups with
$p$ prime, that is, there exists a subgroup
$N$ such that all elements of
$G\setminus N$ are conjugate or inverse-conjugate under
$\mathsf{Aut}(G)$. It is shown that such groups correspond to complete multipartite graphs which are normal edge-transitive Cayley graphs.