Non-inner automorphisms of order p in finite p-groups of coclass 4 and coclass 5

TL;DR

This paper proves the long-standing conjecture that every finite nonabelian p-group has a non-inner automorphism of order p, specifically for groups of coclass 4 and 5, and under certain conditions for odd order groups.

Contribution

It establishes the conjecture for finite p-groups of coclass 4 and 5, and for certain odd order groups with cyclic centers, advancing understanding of automorphisms in p-groups.

Findings

Proved the conjecture for p-groups of coclass 4 and 5.

Extended the conjecture to certain odd order p-groups with specific center conditions.

Confirmed the existence of non-inner automorphisms of order p in these classes.

Abstract

A long-standing conjecture asserts that every finite nonabelian $p$-group has a non-inner automorphism of order $p$. In this paper we prove the conjecture for finite $p$-groups of coclass $4$ and coclass $5$ ($p\ge 5$). We also prove the conjecture for an odd order nonabelian $p$-group $G$ with cyclic center satisfying $Z_3(G)\cap C_G(G^p\gamma_3(G))\le Z(\Phi(G))$.