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

Abstract

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 = p k elements for some prime p > 3 and positive integer k. Upto 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.

A short note on the Wedderburn components of a semisimple finite group algebra

“…Here t is unknown, F q i , for each i, is an unknown finite extension of F q and n ′ i s are also unknowns. In the subject of semisimple group algebras, deducing the WD is a very important and extensively studied reserach problem (see [1,2,3,4,7,12,13,14,15,16,17,18,19]).…”

Wedderburn decomposition of a semisimple group algebra $\mathbb{F}_qG$ from a subalgebra of factor group of $G$

“…But we need the unique choice out of these 4 choices. We want to emphasize the fact that the techniques used in papers [8,9,10] are not useful here to deduce the unique choice from the above-mentioned 4 choices. To see this, one can consider every normal subgroup of G and apply Theorem 2.4 to conclude that no unique choice among the above-mentioned four choices can be obtained.…”

“…In this paper, we continue in the same direction and deduce the Wedderburn decomposition of the semisimple group algebra F q (A 5 C 4 ), where A 5 is an alternating group on 5 symbols, C 4 is a cyclic group of order 4 and q = p k . However, we show that the Wedderburn decomposition of the semisimple group algebra F q (A 5 C 4 ) can not be deduced using the similar techniques of [8,9,10]. Consequently, in this paper, we develop some results with which we show that under certain conditions, the Wedderburn decomposition of any semisimple group algebra F q G can be deduced from the Wedderburn decomposition of F q (G/H), where |H| = 2 and H G. This result would be very helpful in determining the Wedderburn decompositions of many other semisimple group algebras.…”

“…In the recent years, several efforts have been made in order to deduce the Wedderburn decompositions of the semisimple group algebras of non-metabelian groups (cf. [8,9,10,13]).…”