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| Main Authors: | , , |
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| Format: | Preprint |
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2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2603.22185 |
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| _version_ | 1866918404113301504 |
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| author | Bhowmick, Sanjit de la Cruz, Javier Martínez-Moro, Edgar |
| author_facet | Bhowmick, Sanjit de la Cruz, Javier Martínez-Moro, Edgar |
| contents | 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 $\mathbb{F}_\ell^αG$. 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 $\mathbb{F}_\ell^αG$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_22185 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Twisted group algebras of faithful split metacyclic groups $C_p \rtimes C_m$ over finite fields Bhowmick, Sanjit de la Cruz, Javier Martínez-Moro, Edgar Rings and Algebras 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 $\mathbb{F}_\ell^αG$. 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 $\mathbb{F}_\ell^αG$ 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. |
| title | Twisted group algebras of faithful split metacyclic groups $C_p \rtimes C_m$ over finite fields |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2603.22185 |