Finite Exceptional p-Groups of Small Order

Abstract

A finite group is said to be exceptional if its minimal degree of a faithful permutation representation is strictly less than that of one of its factor groups, called a distinguished quotient. It was previously unknown if exceptional p-groups of order less than p ⁶ existed for p an odd prime. The author proved in his M.Sc thesis that there are none of order ≤p ⁴ and gave restrictions on the possible existence of distinguished quotients of exceptional groups of order p ⁵. In this article, an exceptional p-group of order p ⁵ is exhibited for p any odd prime.

Key Words:

• Faithful representation
• Finite exceptional groups
• Minimal degree
• p-groups
• Permutation groups

AMS Subject Classification:

• 20B05
• 20B35