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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.

Relative Commutativity Degree of Nonabelian Metabelian Groups of Order 32