Relative commutativity degree of nonabelian metabelian groups of order 32

Abstract

A metabelian group is a group whose commutator subgroup is abelian. Similarly, a group G is metabelian if and only if there exists an abelian normal subgroup, A, such that the quotient group, G / A , is abelian. The scope of this research is only for nonabelian metabelian groups of order 32. The commutativity degree of a group G is the probability that two elements of the group G (chosen randomly with replacement) commute. This probability can be used to measure how close a group is to be abelian. This concept has been extended to the co-prime probability which is defined as the probability of a random pair of elements x and y in G for which the greatest common divisor for the order of x and order of y is equal to one. Furthermore, the study of relative commutativity degree of a subgroup H of a group G which is the probability of an element in H commutes with an element in G is included in this research. Previous researchers have determined the commutativity degree of nonabelian metabelian groups of order at most 32. Meanwhile, the co-prime probability and the relative commutativity degree of both cyclic and noncyclic subgroups H are obtained for nonabelian metabelian groups of order at most 30. Since there is no nonabelian group of order 31, thus in this research the co-prime probability and the relative commutativity degree of cyclic subgroups for nonabelian metabelian groups of order 32 are determined.