A Survey on the Estimation of Commutativity in Finite Groups

Let G be a finite group and let C - {(x, y) epsilon G x G |xy - yx}. Then Pr(G) = |C|/|G|(2) is the probability that two elements of G, chosen randomly with replacement, commute. This probability is a well known quantity, called commutativity degree of G, and indeed gives us an estimation of commutativity in G. In the last four decades this subject has enjoyed a flourishing development. In this article, we give a brief survey on the development of this subject and then we collect several of our results concerning Pr(G) as well as its various generalizations.