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

TL;DR

The paper addresses the problem of determining the profinite genus of groups of the form $\mathbb{Z}^n\rtimes C_{p^2}$. It develops a framework based on the complete classification of indecomposable $\mathbb{Z}C_{p^2}$-lattices and introduces a natural action of the Galois group $G(p^2)$ on their genus, enabling a reduction of semidirect-product isomorphism to twist-equivalence of underlying lattices. By leveraging ideal-class groups of cyclotomic orders and unit-quotient data $U_t$, the authors derive explicit bounds and exact counts for the genus, depending on the $\mathbb{Z}G$-module structure of the kernel lattice $M$ and its decomposition into indecomposables. The results extend Grunewald–Zalesskii’s genus formulas to the $p^2$-order case, providing a complete description in terms of $G(p^2)$-orbits on $H(\mathbb{Z}[\zeta_{p^2}])$ (and related groups), with special cases for small primes. The methods have potential applicability to other $p$-groups of finite integral representation type and illuminate the role of cyclotomic arithmetic in classifying profinite completions of semidirect products.

Abstract

This paper investigates 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.

Theorems & Definitions (46)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4: Theorem 7.8 in reiner1978 or Theorem 1 in jones
• Lemma 5
• proof
• Lemma 6
• proof
• Lemma 7
• REMARK 8