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

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 $n \geq 3$.