RESEARCH ARTICLE

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

Yuriy Yu. Leshchenko^a, Vitaly I. Sushchansky<sup>b

a</sup> Institute of Physics, Mathematics and Computer Science, Bohdan Khmelnytsky National University of Cherkasy, Shevchenko blvd. 79, Cherkasy, Ukraine, 18031
^b Institute of Mathematics, Silesian University of Technology, ul. Kaszubska 23, Gliwice, Poland, 44-100

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 $n \geq 3$.

Keywords: polynomial ring, unitriangular automorphism, finite field, wreath product, nilpotent group, central series.

MSC: 20D15, 20E22, 20E36, 20F14

Received: 22.04.2014
Revised: 22.04.2014

Language: English

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