Involutive and minimal generating sets of Extended Special Linear group $ES{L_3}(\mathbb{Z})$, $ES{L_2}(\mathbb{Z})$ and formulas of roots in GL$_2$($\mathbb{F}_p$), GL$_2(\mathbb{Z})$ and SL$_3(\mathbb{Z})$ \, \, \, \RomanNumeralCaps{2}

TL;DR

This paper explores the structure of extended special linear groups over integers, introduces new generating sets involving involutions, and derives formulas for roots in various matrix groups, advancing understanding of their algebraic properties.

Contribution

It generalizes the structure of unimodular matrices, proposes an extension of the special linear group, and provides formulas for roots in matrix groups, which were previously unknown or incomplete.

Findings

Identified involutive generating sets for extended special linear groups.

Derived recursive formulas for roots in SL(2, Z).

Found an analytical formula for roots in SL(3, Z).

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.