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

In this paper questions of the width in verbal subgroups of A n is considered. Verbal width by set of squares of alternating group is found, it is happened equals to 2. The criterion of squareness in A n is presented. The necessary and sufficient conditions when an element of alternating group can be presented as a squares of one element are also found by us. Some necessary conditions to an element g ∈ A n being the square in A n are investigated.