A note on normal complement problem

Abstract

In this paper, we study the normal complement problem on semisimple group algebras and modular group algebras FG over a field F of positive characteristic. We provide an infinite class of abelian groups G and Galois fields F that have normal complement in the unit group U(FG) for semisimple group algebras FG. For metacyclic group G of order p1p2, where p1, p2 are distinct primes, we prove that G does not have normal complement in U(FG) for finite semisimple group algebra FG. Finally, we study the normal complement problem for modular group algebras over field of characteristic 2.