Finite Metabelian Group Algebras

Shalini Gupta

doi:10.18052/www.scipress.com/ijpms.17.30

Shalini Gupta

Open Access

https://doi.org/10.18052/www.scipress.com/ijpms.17.30

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Abstract

Given a finite metabelian group G, whose central quotient is abelian (not cyclic) group of order p2, p odd prime, the objective of this paper is to obtain a complete algebraic structure of semisimple group algebra Fq[G] in terms of primitive central idempotents, Wedderburn decomposition and the automorphism group.

Highlights

Let F be a field and G be a finite group such that the group algebra F[G] is semisimple.A fundamental problem in the theory of group algebras is to understand the complete algebraic structure of semisimple group algebra F[G]
Bakshi et al [3] have solved this problem for semisimple finite metabelian group algebras Fq[G], where Fq is a finite of order q and G is a finite metabelian group
They further illustrated their algorithm by explicitly finding a complete set of primitive central idempotents, Wedderburn decomposition and the automorphism group of semisimple group algebra of certain groups whose central quotient is Klein’s four-group

Summary

Introduction

Let F be a field and G be a finite group such that the group algebra F[G] is semisimple.A fundamental problem in the theory of group algebras is to understand the complete algebraic structure of semisimple group algebra F[G]. A complete algebraic structure of semisimple group algebra Fq[G] for some finite groups G, whose central quotient , G Z (G) , is the direct product of two cyclic groups of order p , p odd prime, is obtained. The complete algebraic structure of Fq[G], for group G in the two of the nine classes, is obtained in the present paper .

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Conclusion