Computing irreducible representations of two-generator groups

Abstract

Most of applications of group theory to physical problems are applications of representation theory. Representation theory reduces the properties of groups to numbers. The main part of studying representation theory is to look at irreducible representations of the groups. In this research, irreducible representations of 2- generator p-groups of nilpotency class 2 are studied. The classification of 2-generator p-groups of nilpotency class 2, p an odd prime was introduced by Kappe and Bacon in 1993. For p=2, the classification of 2-generator 2-groups of nilpotency class 2 has been completed by Kappe et al. in 1999. Groups, Algorithms and Programming (GAP) software has been used for the calculations. A character table is formed for each group discussed and the number of irreducible representations for some 2- generator p-groups of nilpotency class 2 are generalized. Since the number of irreducible representations is equal to the number of conjugacy classes, the number of conjugacy classes can be found first. A totally different approach is used. A new structure is revealed such that base cases exist and other groups within the type are central extension. A general formula is found for the first type of group both for p odd and p=2.