For a given positive integer n, the nth commutativity degree of a finite noncommutative semigroup S is defined to be the probability of choosing a pair (x, y) for x, y ∈ S such that xn and y commute in S. If for every elements x and y of an associative algebraic structure (S, . ) there exists a positive integer r such that xy = yrx, then S is called quasi-commutative. Evidently, every abelian group or commutative semigroup is quasi-commutative. In this paper, we study the nth commutativity degree of certain classes of quasi-commutative semigroups. We show that the nth commutativity degree of such structures is greater than 1/2. Finally, we compute the nth commutativity degree of a finite class of non-quasi-commutative semigroups and we conclude that it is less than 1/2.

For a finite group G the commutativity degree denote by d (G) and defind:d (G)=|{(x, y)|x, yÎG, xy=yx}|/|G|2.In [2] authors found commutativity degree for some groups, in this paper we find commutativity degree for a class of groups that have high nilpontencies.

For a finite group G, the probability of two elements of G that commute is the commutativity degree of G denoted by P(G). As a matter of fact, if C = {(a; b) ∈ G×G | ab = ba}, then P(G) = |C|/|G|2 . In this paper, we are going to find few formulas for P(G) independent of  |C|; for some AC-groups, and also in some special cases of finite minimal non-abelian groups. Moreover, the study will present implications for certain qualified finite groups.

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The commutativity degree of a finite non-commutative semigroup S is defined to be the probability of choosing a pair (x, y) of the elements of S such that x commutes with y. Obviously if S is a abelian semigroup, then commutativity degree of S is 1. In this study, we consider semigroups are non-commutative and finite. For a given positive integer n=p^α q^β where p and q are primes (2≤ p.