The exact number of conjugacy classes for 2 - generator p - groups of nilpotency class 2

Abstract

An element x is conjugate to y in a group G if there exists an element g in G such that g-1xg = xg = y. The relation x is conjugate to y is an equivalence relation which induces a partition of G whose elements are called conjugacy classes. The general formula for the exact number of conjugacy classes for nilpotent groups does not exist. Researchers give only the lower bounds for the number of conjugacy classes of nilpotent groups. In this thesis, 2-generator p-groups of nilpotency class 2 (p an odd prime) are considered for their exact number of conjugacy classes. These groups have been classified by Bacon and Kappe in 1993. In 1999, Kappe, Visscher and Sarmin have corrected minor errors on the groups in the classification. Groups, Algorithms, and Programming (GAP) software is used in this research to gain insight into the structure of these groups. There are infinitely many of these groups which are partitioned into three types. For each type, there are infinitely many base groups. New structural results are found such that groups other than base groups are central extensions. As a result of this research, a general formula is derived for the exact number of conjugacy classes for each type of 2-generator p-groups of nilpotency class 2 (p an odd prime)