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Unit groups of the finite group algebras of generalized quaternion groups. (English) Zbl 1454.16042

Let \(F\) be a finite field of size \(q\), where \(q\) is an odd prime power. The goal of this paper is to describe the structure of the group algebra \(FQ_{2^n}\) and its unit group \(U(FQ_{2^n})\), where \(Q_{2^n}\) is the generalised quaternion group of order \(2^n\). Such a description is given in terms of the residue of \(q\) modulo \(2^{n-1}\). In particular, if \(n\leq 7\), then the exact structure of \(U(FQ_{2^n})\) is given for every finite field \(F\) of odd characteristic.

MSC:

16U60 Units, groups of units (associative rings and algebras)
20C05 Group rings of finite groups and their modules (group-theoretic aspects)
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