Group cosets with all elements of equal order

Let $G$ be a finite group and $N$ a proper, nontrivial, normal subgroup of $G$. If, for every element $x$ of $G$ not lying in $N$, the elements in the coset $xN$ all have the same order as $x$, then we say that $(G,N)$ is an {\it{equal order pair}}. This generalizes the concept of a Camina pair, that was introduced by the first author. In the present paper we study several properties of equal order pairs, showing that in many respects they resemble Camina pairs, but with some important differences.