A study of units in group algebras

Abstract

The present thesis mainly concentrates on the problems of determining the structure of the unit groups and unitary subgroups of finite modular group algebras. We obtain the structure of the unit group and the unitary subgroup of the group algebra F D2pm, where D2pm is the dihedral group of order 2p m such that p is an odd prime, m is any positive integer and F is a finite field with characteristic 2. For m = 1, we obtain the structure of the unit group and unitary subgroup of the group algebra F D2p over a finite field F with characteristic p. Further, we study the order of U(F(G n C2n )) in terms of the order of U(F G) for an arbitrary finite group G over a finite field F with characteristic 2. In particular, the order of U(F D2 im) is obtained, where D2 im is the generalized dihedral group of order 2 im such that i ≥ 2 and m is an odd integer. Furthermore, if A is an elementary abelian 2-group, then we obtain the structures of U(F(G × A)) and its unitary subgroup U∗(F(G × A)), where ∗ is the canonical involution of the group algebra F(G × A). In particular, we obtain a set of generators of U∗(F D4m) as well as of U(F D4m). We also study the normal complement problem on semisimple group algebras F G over a field F of positive characteristic. We obtain an infinite class of abelian groups G and Galois fields F that have a normal complement in the unit groups U(F G) of their group algebras F G . Also for a metacyclic group G of order p1p2 with p1, p2 distinct primes, we prove non existence of normal complement in the unit group U(F G) for finite semisimple group algebra F G. Finally, we study this problem for modular group algebras over a field of characteristic 2 and provide some results on symmetric groups and dihedral groups.