  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.
