Probability of having $n^{th}$-roots and n-centrality of two classes of groups

Abstract

In this paper, we consider the finitely 2-generated groups $K(s,l)$ and $G_m$ as follows:<br />$$K(s,l)=langle a,b|ab^s=b^la, ba^s=a^lbrangle,<br />G_m=langle a,b|a^m=b^m=1, {[a,b]}^a=[a,b], {[a,b]}^b=[a,b]rangle$$ <br />and find the explicit formulas for the probability of having nth-roots for them. Also, we investigate integers n for which, these groups are n-central.