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The structure of $\mathbb{Z}_nG$ and its unit group

Jyoti Garg, Sugandha Maheshwary, Himanshu Setia

Abstract

This article determines the structure of the group ring $\mathbb{Z}_nG$, where $G$ is a finite group and $\mathbb{Z}_n$ is the ring of integers modulo $n$, such that $n$ is relatively prime to the order of $G$. The decomposition of $\mathbb{Z}_nG$ is given as a direct sum of matrix rings over Galois rings, thereby extending the structural theory of group rings beyond the classical field setting. We also provide a method to compute a generating set of the unit group $\mathcal{U}(\mathbb{Z}_nG)$, in terms of elementary matrices, using Shoda pair theory. The results are illustrated with examples.

The structure of $\mathbb{Z}_nG$ and its unit group

Abstract

This article determines the structure of the group ring , where is a finite group and is the ring of integers modulo , such that is relatively prime to the order of . The decomposition of is given as a direct sum of matrix rings over Galois rings, thereby extending the structural theory of group rings beyond the classical field setting. We also provide a method to compute a generating set of the unit group , in terms of elementary matrices, using Shoda pair theory. The results are illustrated with examples.

Paper Structure

This paper contains 7 sections, 13 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Notation and Preliminaries
  3. A decomposition of $\mathbb{Z}_{p^r}G,\ r \geq 1$
  4. The unit group of $\mathbb{Z}_{p^r}G$
  5. $\mathcal{U}(\mathbb{Z}_{p^r}G),~G$ cyclic
  6. $\mathcal{U}(\mathbb{Z}_{p^r}G),~G$ strongly monomial
  7. Example

Key Result

Theorem 2.1

BdR07 Let $G$ be a finite group and let $\mathbb{F}_p$ be a field of order $p$ such that $\mathbb{F}_pG$ is semisimple. If $(H, K)$ is a strong Shoda pair of $G$ and $\mathcal{C}\in \mathcal{C}_q(H/K)$, then $e_{\mathcal{C}}(G, H, K)$ is a primitive central idempotent of $\mathbb{F}_pG$ and $\mathb

Theorems & Definitions (26)

  • Theorem 2.1
  • Theorem 3.1
  • proof
  • Remark 3.2
  • Corollary 3.3
  • proof
  • Example 3.4
  • Example 3.5
  • Lemma 4.1
  • proof
  • ...and 16 more