Finding Minimal Permutation Representations of Finite Groups

TL;DR

This paper investigates the structure of minimal permutation representations of finite groups, revealing conditions under which they can be constructed greedily and characterizing their sizes relative to the group order.

Contribution

It introduces a greedy method for constructing minimal permutation representations and establishes size bounds related to the group's order.

Findings

Minimal permutation representations can often be obtained via a greedy construction.

All minimal representations have the same orbit sizes under certain conditions.

Size of minimal faithful G-sets is approximately |G|/m for some integer m, when large enough.

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.