Assume F is a finite field of order and q is an odd prime for which , where and . In this article, we obtain the order of the symmetric and the unitary subgroup of the semisimple group algebra Further, for the extension G of by an abelian group A of order with , we prove that if or and , then G does not have a normal complement in V(FG)

We explore the problem of determining isomorphisms between the twisted complex group algebras of finite p-groups. This problem bears similarity to the classical group algebra isomorphism problem and has been recently examined by Margolis-Schnabel. Our focus lies on a specific invariant, referred to as the generalized corank, which relates to the twisted complex group algebra isomorphism problem. We provide a solution for non-abelian p-groups with generalized corank at most three.

Let G be a finite group and r be a prime divisor of the order of G. An irreducible character of G is said to be quasi r-Steinberg if it is non-zero on every r-regular element of G. A quasi r-Steinberg character of degree |Sylr(G)| is said to be weak r-Steinberg if it vanishes on the r-singular elements of G. In this article, we classify the quasi r-Steinberg cuspidal characters of the general linear group GL(n,q). Then we characterize the quasi r-Steinberg characters of GL(2,q) and GL(3,q). Finally, we obtain a classification of the weak r-Steinberg characters of GL(n,q).