On Orbits of the Ring $Z_n^m$ under the Action of the Group $SL(m,Z_n)$

TL;DR

This paper investigates the orbits formed by the action of the special linear group over the ring of integers modulo n on the m-dimensional vector space, generalizing previous work for prime n and dimension 2.

Contribution

It generalizes the classification of orbits under $SL(m,Z_n)$ action to arbitrary n and dimension m, extending prior results limited to prime n and m=2.

Findings

Determined orbit structure for $SL(m,Z_n)$ acting on $Z_n^m$ for all natural n.

Extended previous results from prime n to arbitrary n.

Provided a comprehensive description of the orbit partition.

Abstract

We consider the action of the finite matrix group $SL(m,Z_n)$ on the ring $Z_n^m$. We determine orbits of this action for n arbitrary natural number. It is a generalization of the task which was studied by A.A. Kirillov for $m=2$ and $n$ prime number.