in this paper, we consider the finitely presented groups $g_{m}$ and $k(s,l)$ as follows;$$g_{m}=langle a,b| a^m=b^m=1,~[a,b]^a=[a,b],~[a,b]^b=[a,b]rangle $$$$k(s,l)=langle a,b|ab^s=b^la,~ba^s=a^lbrangle;$$and find the $n^{th}$-commutativity degree for each of them. also we study the concept of $n$-abelianity on these groups, where $m,n,s$ and $l$ are positive integers, $m,ngeq 2$ and $g.c.d(s,...

‎we consider two commutativity ratios $pr(g)$ and $f(g)$ in a finite group $g$‎ ‎and examine the properties of $g$ when these ratios are `large'‎. ‎we show that‎ ‎if $pr(g) > frac{7}{24}$‎, ‎then $g$ is metabelian and we give threshold‎ ‎results in the cases where $g$ is insoluble and $g'$ is nilpotent‎. ‎we also‎ ‎show that if $f(g) > frac{1}{2}$‎, ‎then $f(g) = frac{n+1}{2n}$‎, ‎for some‎ ‎na...

‎‎for a finite group $g$ and a subgroup $h$ of $g$‎, ‎the relative commutativity degree of $h$ in $g$‎, ‎denoted by $d(h,g)$‎, ‎is the probability that an element of $h$ commutes with an element of $g$‎. ‎let $mathcal{d}(g)={d(h,g):hleq g}$ be the set of all relative commutativity degrees of subgroups of $g$‎. ‎it is shown that a finite group $g$ admits three relative commutativity degrees if a...