On the unit group of a semisimple group algebra and the normal complement problem

Let F be the field with p elements, where p is of the form (2t + 1) for some square free odd integer t. In this article, we obtain the order of the symmetric and the unitary subgroup of U(FCq), where q is a prime divisor of t. Consequently, we resolve the normal complement problem for the modular group algebra of a split extension of Cq by an abelian group of order pm with m >= (q -3), over the field with p elements such that p = (2q+1). Further, we study the normal complement problem in the finite semisimple group algebras of general linear groups.