An extension of commutativity degree of finite groups

Let G be a finite group. The commutativity degree of G, written d(G), is defined as the ratio|{(x, y)|x, y \in G, xy = yx}|/|G|^۲. In this paper, we first extend this concept of finite groups to thecommutativity degree of fuzzy subgroups. Then, by using the numerical solutions of the equation xy − zu \eqiv t(mod n), we give explicit formulas for the commutativity degree of fuzzy subgroups of ۲-generated groups of nilpotency class ۲. Finally we show that this method also works for a large class of finite groups, includingmetabelian groups.

کلیدواژه ها:

Nilpotent groups ، commutativity degree ، fuzzy groups