Exceptional groups of order $p^6$ for primes $p\geq 5$

TL;DR

This paper investigates the structure of exceptional groups of order p^6 for primes p ≥ 5, showing their proportion diminishes asymptotically and identifying a specific count of such groups.

Contribution

It proves the asymptotic rarity of exceptional groups of order p^6 and explicitly counts and conjectures the total number of these groups.

Findings

Proportion of exceptional groups tends to zero as p increases

Identifies exactly (11p+107)/2 exceptional groups for each prime p ≥ 5

Conjectures no additional exceptional groups exist beyond those counted

Abstract

The minimal faithful permutation degree $\mu(G)$ of a finite group $G$ is the least integer $n$ such that $G$ is isomorphic to a subgroup of the symmetric group $S_n$. If $G$ has a normal subgroup $N$ such that $\mu(G/N) > \mu(G)$, then $G$ is exceptional. We prove that the proportion of exceptional groups of order $p^6$ for primes $p \geq 5$ is asymptotically 0. We identify $(11p+107)/2$ such groups and conjecture that there are no others.