Characterisation of Unit Group of Semisimple Group Algebras of Generalised Heisenberg Group and Some Class of Non Metabelian Groups

Abstract

This thesis addresses the challenges encountered in the study of the unit group of newlinesemisimple group algebras. This work systematically explores these challenges by focusing on newlinespecific types of groups and their associated group algebras. Our research particularly examines newlinethe unit groups of group algebras corresponding to several groups, including the Heisenberg newlinegroup in higher dimensions, all symmetric groups, and a selection of non-metabelian groups of newlineorders 128, 144, 150, 160, 162, 168, and 180. The choice of these groups stems from their diverse newlinealgebraic properties and the unique challenges each presents when investigating their unit group newlineof the group algebras. newlineA significant finding of this thesis is the identification of the inherent complexities and newlineobstacles in proving the unit group structure of each group algebra. We demonstrate that the newlinemethod applied to one group algebra is not necessarily effective for others. This variation in newlineapproach is largely due to the distinct algebraic properties of the groups under consideration. For newlineexample, while the unit group of the group algebra of a symmetric group might be approached newlinethrough a certain combinatorial aspect, the same framework might fail or require significant newlinemodification when applied to the group algebra of a non-metabelian group of a similar order newline