Automorphisms of small prime power groups

TL;DR

This paper investigates the proportion of small prime power groups with automorphism groups that are also prime power groups, showing that this ratio remains bounded away from 1 for large primes and providing data for small primes.

Contribution

It proves that for groups of order up to p^7, the ratio of groups with p-group automorphism groups stays bounded away from 1 as p increases, and supplies empirical data for small primes.

Findings

The ratio g(p,n)/f(p,n) is bounded away from 1 for n ≤ 7 as p grows.

Data on groups with automorphism groups of prime power order for primes ≤ 11.

Provides bounds and empirical data on automorphism group structures.

Abstract

If $f(p, n)$ is the number of pairwise nonisomorphic groups of order $p^n$, and $g(p,n)$ is the number of groups of order $p^n$ whose automorphism group is a $p$-group, then, for $n \leq 7$, we prove that the ratio $g(p,n)/f (p,n)$ is bounded away from 1 as the prime $p$ grows to infinity. In addition, we provide some data on the number of groups whose automorphism group is a group of prime power order, for primes no larger than 11.