The commutativity degree of some nonabelian two-groups with a cyclic subgroup of index four

Abstract

The determination of the abelianness of a finite group has been introduced for symmetric groups, finite groups and finite rings in the last fifty years. The abelianness of a group or known as the commutativity degree of a group is defined as the probability that a random pair of elements in the group commute. The basic probability theory is used in studying its connection with group theory. The aim of this study is to determine the commutativity degree of some nonabelian 2-groups with a cyclic subgroup of index 4. Two approaches have been used to calculate the commutativity degree of those groups. The first approach is by using a formula involving the number of conjugacy classes and the second approach is by using the Cayley Table method. In this thesis, some basic concepts of the commutativity degree of finite group are first reviewed then the computation of the commutativity degree of nonabelian 2-groups with a cyclic subgroup of index 4 of order 16 and 32 are done.