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.
