Rachel Camina

A new generalisation of Camina pairs

Let G be a finite group and N a nontrivial, proper, normal subgroup of G. In 1978 Alan Camina considered pairs (G,N) satisfying the following property: each coset xN ≠ N is contained in a single conjugacy class (necessarily that of x). His aim was to characterise Frobenius groups. The paper spawned much research and the pairs became known as Camina pairs. We relax this condition and just insist each coset contains elements of the same order. We show that in many ways these new pairs resemble Camina pairs, but with some important differences.
This is joint work with Alan Camina, Mark L Lewis, Emanuele Pacifici, Lucia Sanus and Marco Vergani.