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

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 ${\mathbb{Z}}_n^m={\mathbb{Z}}_n\times {\mathbb{Z}}_n\times ...\times {\mathbb{Z}}_n$ 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 $n\ge 2$ is a group under matrix multiplication modulo $n$, on the ring ${\mathbb{Z}}_n^m$ as a right multiplication of a row vector of ${\mathbb{Z}}_n^m$ by a matrix of $SL(m,{\mathbb{Z}}_n)$ to determine the orbits of the ring ${\mathbb{Z}}_n^m$. This work is an extension of [1]

Some improvements in this note are in process