Some automorphism groups of finite p-groups

TL;DR

This paper proves Berkovich's conjecture for odd p-groups of coclass 2, showing they have non-inner automorphisms of order p, and presents related independent results.

Contribution

It establishes the conjecture for a specific class of finite p-groups and introduces new related automorphism results.

Findings

Proved the conjecture for p-groups of coclass 2 with odd p

Identified conditions for existence of non-inner automorphisms of order p

Presented additional automorphism-related results

Abstract

A conjecture of Berkovich asserts that every non-simple finite p-group has a non-inner automorphism of order p. This conjecture is far from being proved despite the great effort devoted to it. In this paper we prove it for p-groups of coclass 2, provided that p is odd. Some related results are also proved, and may be considered as interesting independently.