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

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.