‎suppose $n$ is a fixed positive integer‎. ‎we introduce the relative n-th non-commuting graph $gamma^{n} _{h,g}$‎, ‎associated to the non-abelian subgroup $h$ of group $g$‎. ‎the vertex set is $gsetminus c^n_{h,g}$ in which $c^n_{h,g} = {xin g‎ : ‎[x,y^{n}]=1 mbox{~and~} [x^{n},y]=1mbox{~for~all~} yin h}$‎. ‎moreover‎, ‎${x,y}$ is an edge if $x$ or $y$ belong to $h$ and $xy^{n}eq y^{n}x$ or $x...

The second author introduced notions of weak permutablity and commutativity between groups and proved the finiteness of a group generated by two weakly permutable finite subgroups. Two groups H, K weakly commute provided there exists a bijection f : H → K which fixes the identity and such that h commutes with its image h for all h ∈ H. The present paper gives support to conjectures about the ni...

A group G is called power commutative, or a PC-group, if [xm, yn] = 1 implies [x, y] = 1 for all x, y ∈G such that xm = 1, yn = 1. So power-commutative groups are those groups in which commutativity of nontrivial powers of two elements implies commutativity of the two elements. Clearly, G is a PC-group if and only if CG(x) = CG(x) for all x ∈ G and all integers n such that xn = 1. Obvious examp...