A note on the action of $SL(m, \mathbb{Z}_n)$ on the ring $\mathbb{Z}^m_n$

TL;DR

This paper investigates how the special linear group over modular integers acts on the ring of m-tuples over the same modular ring, extending previous work on orbit structures under this group action.

Contribution

It describes the action of $SL(m, bZ_n)$ on $bZ_n^m$ for composite $n$, extending prior results that focused on prime moduli.

Findings

Characterization of orbits under the group action

Extension of previous prime modulus results to composite moduli

Description of the group action as right multiplication

Abstract

We know that $\mathbb{Z}_n$ is a finite field for a prime number $n$. Let $m,n$ be arbitrary natural numbers and let $\mathbb{Z}^m_n= \mathbb{Z}_n \times\mathbb{Z}_n\times...\times\mathbb{Z}_n$ be the Cartesian product of $m$ rings $\mathbb{Z}_n$. In this note, we present the action of $SL(m, \mathbb{Z}_n)=\{A \in \mathbb{Z}^{m,m}_{n} : det A \equiv 1 (mod\simn)\}$, where $SL(m, \mathbb{Z}_n)$ for $n\geq 2$ is a group under matrix multiplication modulo $n$, on the ring $\mathbb{Z}^m_n$ as a right multiplication of a row vector of $\mathbb{Z}^m_n$ by a matrix of $SL(m, \mathbb{Z}_n)$ to determine the orbits of the ring $\mathbb{Z}^m_n$. This work is an extension of [1]