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Units in \(\mathbb F_{q^k}(C_p\rtimes_rC_q)\). (English) Zbl 1320.16020
Summary: Let \(q\) be a prime, \(\mathbb F_{q^k}\) be a finite field having \(q^k\) elements and \(C_p\rtimes_rC_q\) be a group with presentation \(\langle a,b\mid a^n,\;b^q,\;b^{-1}ab=a^r\rangle\), where \((n,rq)=1\) and \(q\) is the multiplicative order of \(r\) modulo \(n\). In this paper, we address the problem of computing the Wedderburn decomposition of the group algebra \(\mathbb F_{q^k}(C_p\rtimes_rC_q)\) modulo its Jacobson radical. As a consequence, the structure of the unit group of \(\mathbb F_{q^k}(C_p\rtimes_rC_q)\) is obtained when \(p\) is a prime different from \(q\).
MSC:
16U60 Units, groups of units (associative rings and algebras)
16S34 Group rings
20C05 Group rings of finite groups and their modules (group-theoretic aspects)
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