Square root of an element in $PSL_2(\mathbb{F}_p)$, $SL_2(\mathbb{F}_p)$, $GL_2(\mathbb{F}_p)$ and $A_n$. Verbal width by set of squares in alternating group $A_n$ and Mathieu groups

TL;DR

This paper develops criteria for determining when elements in groups like $PSL_2(F_p)$, $SL_2(F_p)$, $GL_2(F_p)$, and $A_n$ are perfect squares, extending previous work to finite fields including characteristic 2.

Contribution

It introduces new necessary and sufficient conditions for element squareness in these groups, especially over fields of characteristic 2, and analyzes verbal width by squares in alternating groups and Mathieu groups.

Findings

Criteria for square roots in $SL_2(F_p)$ and $PSL_2(F_p)$ are established.

Conditions for elements in $A_n$ and $GL_2(F_p)$ to be squares are derived.

The work extends existing results to fields of characteristic 2, including $F_2$ and $F_{2^n}$.

Abstract

The problems of square root from group element existing in $SL_2(F_p)$, $PSL_2(F_p)$ and $GL_2(F_p)$ were solved. The similar goal of root finding was reached in the GM algorithm adjoining an $n$-th root of a generator results in a discrete group for group $PSL(2,R)$, but we consider this question over finite field $F_p$. Well known the Cayley-Hamilton method \cite{Pell} for computing the square roots of the matrix $M^n$ can give answer of square roots existing over finite field only after computation of $det M^n$ and some real Pell-Lucas numbers by using Bine formula. Over method gives answer about existing $\sqrt{ M^n}$ without exponents $M$ to $n$-th power. We use only trace of $M$ or only eigenvalues of $M$. In paper "Computing n-th roots in SL2 and Fibonacci polynomials" it was only the Anisotropic case of group $SL_1(Q)$ solved, where $Q$ is a quaternion division algebra over $k$ was considered. The authors of \cite{Amit} considered criterion to be square only for case $F_p$ is a field of characteristic not equal 2. We solve this problem even for fields $F_2$ and $F_{2^n}$. The criterion to $g \in SL_2 (F_2)$ be square in $SL_2(F_2)$ was not found by them what was declared in a separate sentence. The criterion of squareness in $A_n$ is presented. The necessary and sufficient conditions when an element of alternating group $g A_n$ and $GL_2(F_p)$ as well as for $SL_2(F_p)$ can be presented as a squares of one element are also found by us. Some necessary conditions to an element $g\in A_n$ being the square in $A_n$ are investigated. The criterion square root of an element existing in $PSL_2(\mathbb{F}_p)$ is found. The criterion of existing an element square root in $PSL_2(\mathbb{F}_p)$ is found.