Unit group of some finite semisimple group algebras

Abstract

We provide the structure of the unit group of $${\mathbb {F}}_{p^k}S_n$$ F p k S n , where $$p>n$$ p > n is a prime and $$S_n$$ S n denotes the symmetric group on n letters. We also provide the complete characterization of the unit group of the group algebra $${\mathbb {F}}_{p^k}A_6$$ F p k A 6 for $$p\ge 7$$ p ≥ 7 , where $$A_6$$ A 6 is the alternating group on 6 letters.

Units of the semisimple group algebras FqSL(2,8) and FqSL(2,9)

“…The unit groups of the group algebras of non-metabelian groups up to order 72 were described by Mittal et al in [16]. Further, Sharma et al determined the unit group of the semisimple group algebra (SGA) of the respective groups SL(2, 3) (special linear group over the finite field of 3 elements) and SL (2,5) (See [11] and [19]). In continuation, Sivaranjani et al [21] determined the unit group of the SGA of the group SL (2,7).…”

“…Further, Sharma et al determined the unit group of the semisimple group algebra (SGA) of the respective groups SL(2, 3) (special linear group over the finite field of 3 elements) and SL (2,5) (See [11] and [19]). In continuation, Sivaranjani et al [21] determined the unit group of the SGA of the group SL (2,7). In addition, Arvind et al [1] investigated the unit group of the SGA of the group SL(3, Z 2 ).…”

“…In continuation, Sivaranjani et al [21] determined the unit group of the SGA of the group SL (2,7). In addition, Arvind et al [1] investigated the unit group of the SGA of the group SL(3, Z 2 ). The main objective of this paper is to derive the unit group of the SGA of the groups SL (2,8) and SL (2,9), respectively.…”

“…In addition, Arvind et al [1] investigated the unit group of the SGA of the group SL(3, Z 2 ). The main objective of this paper is to derive the unit group of the SGA of the groups SL (2,8) and SL (2,9), respectively. We notice that the difficulty of exactly identifying the unit group of the SGA increases as the size of the group increases.…”

“…The prerequisites for the article are covered in Section 2. In sections 3 and 4, we deduce the unit group of the group algebras F q SL(2, 8) and F q SL (2,9) in the form of theorems 3.1 and 4.1, respectively. Section 5 concludes the paper.…”