The relative commutativity degree and sub-multiplicative degree for noncyclic subgroups of some nonabelian metabelian groups

Abstract

A metabelian group is a group G that has at least an abelian normal subgroup N such that the quotient group G/n is also abelian. The concept of commutativity degree plays an im portant role in determining the abelianness of the group. This concept has been extended to the relative commutativity degree of a subgroup H of a group G which is defined as the probability that an element of H commutes with an element of G. This notion is further extended to the notion of the multiplicative degree of a group G which is defined as the probability that the product of a pair of elements chosen randomly from a group G is in the given subgroup of H . By using those two definitions with an assistance from Groups, Algorithms and Programm ing and Maple software, the relative commutativity degree and sub-multiplicative degree for noncyclic subgroups of nonabelian metabelian groups of order less than 24 and dihedral groups of order at most 24 are determined in this dissertation.