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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 wasreached in the GM algorithm adjoining an $n$-th root of a generator results ina discrete group for group $PSL(2,R)$, but we consider this question overfinite field $F_p$. Well known the Cayley-Hamilton method \cite{Pell} forcomputing the square roots of the matrix $M^n$ can give answer of square rootsexisting over finite field only after computation of $det M^n$ and some realPell-Lucas numbers by using Bine formula. Over method gives answer aboutexisting $\sqrt{ M^n}$ without exponents $M$ to $n$-th power. We use only traceof $M$ or only eigenvalues of $M$.  In paper "Computing n-th roots in SL2 and Fibonacci polynomials" it was onlythe Anisotropic case of group $SL_1(Q)$ solved, where $Q$ is a quaterniondivision algebra over $k$ was considered. The authors of \cite{Amit} consideredcriterion to be square only for case $F_p$ is a field of characteristic notequal 2. We solve this problem even for fields $F_2$ and $F_{2^n}$. Thecriterion to $g \in SL_2 (F_2)$ be square in $SL_2(F_2)$ was not found by themwhat was declared in a separate sentence.  The criterion of squareness in $A_n$ is presented. The necessary andsufficient 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 oneelement are also found by us. Some necessary conditions to an element $g\inA_n$ being the square in $A_n$ are investigated. The criterion square root ofan element existing in $PSL_2(\mathbb{F}_p)$ is found. The criterion ofexisting an element square root in $PSL_2(\mathbb{F}_p)$ is found.

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