On unit group of finite semisimple group algebras of non-metabelian groups of order 108

Gaurav Mi̇ttal,Rajendra K Sharma

doi:10.13069/jacodesmath.935938

Abstract

In this paper, we characterize the unit groups of semisimple group algebras $\mathbb{F}_qG$ of non-metabelian groups of order $108$, where $F_q$ is a field with $q=p^k$ elements for some prime $p > 3$ and positive integer $k$. Up to isomorphism, there are $45$ groups of order $108$ but only $4$ of them are non-metabelian. We consider all the non-metabelian groups of order $108$ and find the Wedderburn decomposition of their semisimple group algebras. And as a by-product obtain the unit groups.

Highlights

Introduction8(2) (2021) 59–71 deduce that there are non-metabelian groups of order of 48, 54, 60, 72, 108 etc
In this paper, we characterize the unit groups of semisimple group algebras FqG of non-metabelian groups of order 108, where Fq is a field with q = pk elements for some prime p > 3 and positive integer k
Let Fq denote a finite field with q = pk elements for odd prime p > 3, G be a finite group and FqG be the group algebra

Summary

Introduction

8(2) (2021) 59–71 deduce that there are non-metabelian groups of order of 48, 54, 60, 72, 108 etc. The unit groups of the group algebras of non-metabelian groups up to order 72 have been discussed in [10, 12]. The unit group of the semisimple group algebra of the non-metabelian group SL(2, 5) has been discussed in [15]. The main motive of this paper is to characterize the unit groups of FqG, where G represents a non-metabelian group of order 108. We deduce the Wedderburn decomposition of group algebras of all the 4 non-metabelian groups and characterize the respective unit groups. Our main results on the characterization of the unit groups are presented in third section and the last section includes some discussion

Preliminaries

Unit group of FqG where G is a non-metabelian group of order 108

Discussion