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

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.