On minimal faithful permutation representations of finite groups

Abstract

The minimal (faithful) degree μ( G ) of a finite group G is the least positive integer n such that G ≲ S n . Clearly if H ≤ G then μ( H ) ≤ μ( G ). However if N ◃ G then it is possible for μ( G / N ) to be greater than μ( G ); such groups G are here called exceptional. Properties of exceptional groups are investigated and several families of exceptional groups are given. For example it is shown that the smallest exceptional groups have order 32.