On the relative commutativity degree of a subgroup of a finite group

The aim of this article is to give a generalization of the concept of commutativity degree of a finite group G (denoted by d(G)), to the concept of relative commutativity degree of a subgroup H of a group G (denoted by d(H, G)). We shall state some results concerning the new concept which are mostly new or improvements of known results given in Gustafson (1973) and Moghaddam et at (2005). Moreover, we shall define;the relative nth nilpotency degree of a subgroup of a group and give some results concerning this at the end of the article.