Minimal Permutation Representations of Nilpotent Groups

Abstract

A minimal permutation representation of a finite group $G$ is a faithful $G$-set with the smallest possible size. We study the structure of such representations and show that for certain groups they may be obtained by a greedy construction. In these situations (except when central involutions intervene) all minimal permutation representations have the same set of orbit sizes. Using the same ideas, we also show that if the size $d(G)$ of a minimal faithful $G$-set is at least $c|G|$ for some $c>0$, then $d(G) = |G|/m + O(1)$ for an integer $m$, with the implied constant depending on $c$.