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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)
##### Keywords:
unit groups; group algebras; Wedderburn decompositions
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