The profinite genus of the groups $\mathbb{Z}^n\rtimes C_{p^2}$

TL;DR

This paper provides a formula for the profinite genus of certain semidirect product groups, specifically those involving $Z^n$ and cyclic groups of order $p^2$, advancing the classification of these groups.

Contribution

It completes the calculation of the profinite genus for groups of the form $Z^n times C_{p^2}$, extending to all such semidirect products with finite $p$-groups of finite representation type.

Findings

Derived a formula for the profinite genus of $Z^n times C_{p^2}$ groups.

Extended the calculation to all $Z^n times G$ where $G$ is a finite $p$-group of finite representation type.

Contributed to the classification of these groups based on their profinite properties.

Abstract

A formula is given for the profinite genus of groups of the form $\mathbb{Z}^n \rtimes C_{p^2}$, completing the calculation of the size of the genus of semidirect products of the form $\mathbb{Z}^n \rtimes G$ where $G$ is a finite $p$-group of finite integral representation type.