Unit group of group algebra

Abstract

The present thesis primarily focusses on the study of structure of unit group of group algebras. We aim at obtaining information about conjugacy classes in the unit group of various modular group algebras. Further, we employ the knowledge of cardinality of certain disjoint conjugacy classes in the normalized unit group to resolve the normal complement problem. In this direction, we start with obtaining size of each conjugacy class in normalized unit group of finite modular group algebra of the group G = H A; where H is a non-abelian group of order 8 and A is an elementary abelian 2-group. We obtain representatives of conjugacy classes in V (FH) and give a partition of V (FH) into conjugacy classes. Moreover, we provide a normal complement of these groups in the normalized unit group V (FG) as well as in the unitary subgroup V (FG); where F is the field with 2 elements. Further, we study the normal complement problem for modular group algebras of some split metabelian p-groups and dihedral groups of order 2pm. First we affirmatively solve the normal complement problem for the group algebras of finite split metabelian p-groups of exponent p over the field with p elements, where p is an odd prime. Then we settle the normal complement problem for group algebras FD2pm; where F is a finite field of characteristic p: We continue the study with group algebras FG; where F is a finite field of characteristic p such that 3 j p􀀀1 and G = AoC3 for a finite abelian p-group A:With the information of certain sets of conjugacy classes that we obtain, we are able to deduce that G does not have a normal complement in V (FG): We also give order and structure of the unitary subgroup V (FG): For an odd prime p; we obtain a relation between the cardinality of conjugacy class of any element of a finite p-group G in its normalized unit group V (FG) and the unitary subgroup V (FG); over a finite field F of characteristic p: Finally, we provide the class length of elements of G in V (FG); where G is a finite p-group of nilpotency class 2 or G is of nilpotency class 3 such that jGj p5: