Twisted group algebras of faithful split metacyclic groups $C_p \rtimes C_m$ over finite fields

TL;DR

This paper develops a comprehensive theory for twisted group algebras of faithful split metacyclic groups over finite fields, including cohomology, idempotents, Wedderburn decomposition, and projective representations.

Contribution

It provides explicit descriptions of the cohomology, primitive central idempotents, algebra decomposition, and projective representations for these twisted group algebras.

Findings

Second cohomology group is isomorphic to _^ imes/(_^ imes)^m.

Complete Wedderburn decomposition into matrix algebras over explicit finite fields.

Determination of all irreducible projective representations over _.

Abstract

Let $\mathbb{F}_\ell$ be a finite field with $\ell$ elements and let $G = C_p \rtimes C_m$ be a faithful split metacyclic group. In this paper, we develop a complete theory for the twisted group algebra $\mathbb{F}_\ell^\alpha G$. Using the Lyndon--Hochschild--Serre spectral sequence, we prove that the second cohomology group of $G$ is isomorphic to $\mathbb{F}_\ell^\times/(\mathbb{F}_\ell^\times)^m$, and we show that all twisting occurs only on the $C_m$ factor. We determine the primitive central idempotents by analyzing the combined action of the Frobenius automorphism and the group action on the character group of $C_p$. Using crossed product theory and the structure of finite fields, we obtain the complete Wedderburn decomposition of $\mathbb{F}_\ell^\alpha G$ into matrix algebras over explicitly determined fields $\mathbb{F}_{\ell^{d_j}}$. Finally, the irreducible projective representations of $G$ over $\mathbb{F}_\ell$ are also determined.