On the group of unitriangular automorphisms of the polynomial ring in two variables over a finite field

Abstract

The group \(U\!J_2(\mathbb{F}_q)\) of unitriangular automorphisms of the polynomial ring in two variables over a finite field \(\mathbb{F}_q\), \(q=p^m\), is studied. We proved that \(U\!J_2(\mathbb{F}_q)\) is isomorphic to a standard wreath product of elementary Abelian \(p\)-groups. Using wreath product representation we proved that the nilpotency class of \(U\!J_2(\mathbb{F}_q)\) is \(c=m(p-1)+1\) and the \((k+1)\)th term of the lower central series of this group coincides with the \((c-k)\)th term of its upper central series. Also we showed that \(U\!J_n(\mathbb{F}_q)\) is not nilpotent if \(n \geq 3\).