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

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.