Wedderburn Decomposition and Idempotents of some finite metacyclic group algebras

TL;DR

This paper explicitly decomposes certain finite metacyclic group algebras into simple components and constructs their idempotents, enhancing understanding of their algebraic structure.

Contribution

It provides an explicit Wedderburn decomposition and constructs all central and left idempotents for specific metacyclic group algebras.

Findings

Explicit Wedderburn decomposition of the algebra

Complete set of central and left idempotents constructed

Applicable to groups with cyclic subgroups of index 2

Abstract

In this article, we show explicitly the Wedderburn decomposition of the metacyclic group algebra $\mathbb F_qG$, where $G$ has a cyclic subgroup of index 2 and $\gcd(|G|,q)=1$. We also construct the complete set of central and left idempotents of these group algebras.