Involutive minimal generating sets with two commuting involutions of Extended Special Linear group $ES{L_3}(\mathbb{Z})$, $ES{L_2}(\mathbb{Z})$ and analytical and recursive formulas of roots in SL$_3(\mathbb{F}_p)$ \RomanNumeralCaps{3}

Abstract

In this research we continue our previous investigation of wreath product normal structure \cite{SkuESL}. We generalize the group of unimodular matrices \cite{Amit} and find its structure. For this goal we propose one extension of the special linear group. Groups generated by three involutions, two of which are permutable, have long been of interest in the theory of matrix groups \cite{Maz}, for instance such generating set was researched for $S{{L}_{2}}({{\mathbb{Z}+ i\mathbb{Z}}})$. But for size of matrix 3 on 3 this is imposable for some groups. We research this question for $ES{{L}_{3}}({{\mathbb{Z}}})$. An analytical formula of root in $SL(3, \mathbb{Z}$) is found, recursive formula for $n$-th power root in $SL(2, \mathbb{Z}$) is found too.