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

TL;DR

This work addresses the existence and frequency of exceptional groups of order $p^6$ for primes $p \ge 5$, where exceptional means $\mu(G/N) > \mu(G)$ for some normal subgroup $N$. The authors develop an upper bound on the number of such groups via an in-depth analysis of isoclinism families and quotient structures, while also producing a concrete candidate list of exceptional groups whose size matches a derived lower bound. They show the proportion of exceptional groups among all groups of order $p^6$ is asymptotically $0$ and identify exactly $(11p+107)/2$ exceptional groups, conjecturing this to be the complete set for all $p \ge 5$. The results are supported by explicit computations for small primes and are accompanied by publicly accessible data and presentations to enable verification and further study. The findings refine our understanding of faithful permutation representations of $p$-groups and suggest exceptional behavior is rare at this order.

Abstract

The minimal faithful permutation degree $μ(G)$ 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 $μ(G/N) > μ(G)$, 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.

Theorems & Definitions (22)

• Theorem 1.1
• Lemma 3.1
• proof
• Corollary 3.2
• proof
• Lemma 3.3
• proof
• Lemma 3.4
• proof
• Lemma 3.5