Assume F is a finite field of order pf and q is an odd prime for which pf-1=sqm, where m≥1 and (s,q)=1. In this article, we obtain the order of the symmetric and the unitary subgroup of the semisimple group algebra FCq. Further, for the extension G of Cq=⟨b⟩ by an abelian group A of order pn with CA(b)={e}, we prove that if m>1, or (s+1)≥q and 2n≥f(q-1), then G does not have a normal complement in V(FG).

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 is said to be weak r-Steinberg if it vanishes on the r-singular elements of In this article, we classify the quasi r-Steinberg cuspidal characters of the general linear group Then we characterize the quasi r-Steinberg characters of and Finally, we obtain a classification of the weak r-Steinberg characters of

Let n be an integer co-prime to 3 and let Zn be the ring of integers modulo n. In this article, we study the structure and generators of the unit group of ZnC3. Further, if Tm denotes the elementary abelian 3-group of order 3m, then we provide the structure of U(ZnTm). We also solve the normal complement problem in each case.

In this article, 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 F be the field with p elements, where p is of the form (2 t+ 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= (2 q+ 1). Further, we study the normal complement problem in the finite semisimple group algebras of general linear groups.

Let p and q be odd primes such that (Formula presented.) Let F be the field with p elements and (Formula presented.) be a group, where A is an abelian group of order (Formula presented.) In this article, we prove that if (Formula presented.) then G does not have a normal complement in (Formula presented.) Further, for any integer (Formula presented.) we prove that if F is a finite field such that (Formula presented.) then (Formula presented.) and (Formula presented.) do not have a normal complement in (Formula presented.) and (Formula presented.) respectively.

In this paper, we obtain the structure of the normalized unit group V (F[x]G) of the modular group algebra F[x]G, where G is a finite abelian group and F[x] is the univariate polynomial ring over a finite field F of characteristic p.

Let p be an odd prime and G be a finite split metabelian p-group of exponent p. In this article, we obtain a normal complement of G in (Formula presented.) where F is the field with p elements. Further, assume that (Formula presented.) where A is a finite abelian p-group and (Formula presented.) If F is any finite field of characteristic p, then we prove that G does not have a normal complement in (Formula presented.) and obtain the structure of the unitary subgroup (Formula presented.) Communicated by Sudarshan Kumar Sehgal.

Let G= H× A be a finite 2-group, where H is a non-abelian group of order 8 and A is an elementary abelian 2-group. We obtain a normal complement of G in the normalized unit group V(FG) and in the unitary subgroup V∗(FG) over the field F with 2 elements. Further, for a finite field F of characteristic 2, we derive class size of elements of V(FG). Moreover, we provide class representatives of V∗(FH).

Let F be a finite field of characteristic p > 0. In this article, we obtain a relation between the class length of elements of a finite p-group G in the normalized unit group V (F G) and its unitary subgroup V*(F G), when p is an odd prime. We also provide the size of the conjugacy class of non-central elements of a group G in V (F G), where either G is any finite p-group with nilpotency class 2 or G is a p-group with nilpotency class 3 such that |G| ≤ p5.

In this article, we discuss the normal complement problem for metacyclic groups in modular group algebras. If F is the field with p elements and G is a finite split metacyclic p-group of nilpotency class 2, then we prove that G has a normal complement in U(FG) For a finite field F of characteristic p, where p is an odd prime, we prove that D2pm has a normal complement in U(FD2pm) if and only if p = 3 and |F|=3.