Finite p-groups with small automorphism group

TL;DR

This paper constructs specific finite p-groups demonstrating that their automorphism groups can be arbitrarily small relative to the groups themselves, disproving a longstanding conjecture about automorphism divisibility.

Contribution

It provides a family of finite p-groups with automorphism groups much smaller than the groups, challenging previous assumptions about automorphism divisibility.

Findings

Constructed infinite family of p-groups with automorphism groups much smaller than the groups

Disproved the conjecture that group order divides automorphism group order for all non-abelian p-groups

Showed that the ratio of automorphism group size to group size can tend to zero

Abstract

For each prime $p$ we construct a family $\{G_i\}$ of finite $p$-groups such that $|\Aut (G_i)|/|G_i|$ goes to $0$, as $i$ goes to infinity. This disproves a well-known conjecture that $|G|$ divides $|\Aut(G)|$ for every non-abelian finite $p$-group $G$.