Let G be a nite group. Then the commutativity degree of G which is denoted by d(G) is the probability that a randomly chosen pair of elements in a group G commutes. In this talk, I will give some upper and lower bounds for d(G) and the exact evaluation for some known groups. Furthermore, I will state some generalizations of commutativ- ity degree to the relative commutativity degree, n-th nilpotency degree, relative n-th nilpotency degree and n-th commutativity degree. Some results and conjectures will be also given in this article.

Keywords