The commutativity degree of all nonabelian metabelian groups of order at most 24

Abstract

A metabelian group is a group whose commutator subgroup is abelian. Equivalently, 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. Meanwhile, the commutativity degree can be viewed as the probability that two elements in a group commute, denoted by P(G) . The main objective of this research is to compute the commutativity degree of all metabelian groups of order at most 24. Some basic concepts related with P(G) will first be presented. Two approaches have been used to compute P(G), where G is a metabelian group of order at most 24, namely the 0-1 Table and the Conjugacy Class Method. A software named Groups, Algorithms and Programming (GAP) have been used to facilitate the computations of the commutativity degree.