On $\CYRSH$-rigidity of groups of order $p^6$

TL;DR

This paper computes the size of the group of class-preserving outer automorphisms for all groups of order p^6 and explores implications for the Bogomolov multiplier in the context of YRSH-rigidity.

Contribution

It provides a complete calculation of class-preserving outer automorphisms for groups of order p^6 and links YRSH-rigidity to the vanishing of the Bogomolov multiplier.

Findings

|Out_c(G)| computed for all groups of order p^6

YRSH-rigid groups of order p^6 have zero Bogomolov multiplier

Establishes a connection between YRSH-rigidity and group cohomology

Abstract

Let $G$ be a group and $Out_c(G)$ be the group of its class-preserving outer automorphisms. We compute $|Out_c(G)|$ for all the group $G$ of order $p^6$, where $p$ is an odd prime. As an application, we observe that if $G$ is a $\CYRSH$-rigid group of order $p^6$, then it's Bogomolov multiplier $B_0(G)$ is zero.