On exceptional groups of order p^5

TL;DR

This paper classifies all pairs of groups of order p^5 with distinguished quotients for odd primes p, revealing that as p grows, about half of such groups have exceptional quotients, highlighting a significant structural property.

Contribution

It provides a complete classification of pairs (G,Q) for groups of order p^5 with distinguished quotients when p is odd, extending previous work and establishing asymptotic behavior.

Findings

As p increases, the proportion of groups with exceptional quotients approaches 1/2.

Complete classification of pairs (G,Q) for groups of order p^5 with distinguished quotients.

The case p=2 was previously addressed by Easdown and Praeger.

Abstract

A finite group G is exceptional if it has a quotient Q whose minimal faithful permutation degree is greater than that of G. We say that Q is a distinguished quotient. The smallest examples of exceptional p-groups have order p^5. For an odd prime p, we classify all pairs (G,Q) where G has order p^5 and Q is a distinguished quotient. (The case p=2 has already been treated by Easdown and Praeger.) We establish the striking asymptotic result that as p increases, the proportion of groups of order p^5 with at least one exceptional quotient tends to 1/2.