Finding Minimal Permutation Representations of Finite Groups

A minimal permutation representation of a finite group G is a faithful G-set with the smallest possible cardinality. We study the structure of such representations and show that for most groups they may be obtained by a greedy construction. It follows that whenever the algorithm works (except when central involutions intervene) all minimal permutation representations have the same set of orbit cardinalities. 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 at most on c.