Relative elementary abelian groups and a class of edge-transitive cayley graphs

© 2015 Australian Mathematical Publishing Association Inc. 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 GnN are conjugate or inverse-conjugate under Aut(G). It is shown that such groups correspond to complete multipartite graphs which are normal edge-transitive Cayley graphs.