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On the normal complement problem for finite group algebras of abelian-by-cyclic groupsUnitary subgroup and normal complement problemA. Herman and S. Kaur
被引：0
作者：
Allen Herman
[1
]
Surinder Kaur
[2
]
机构：
[1] University of Regina,Department of Mathematics and Statistics
[2] SRM University-AP,Department of Mathematics, School of Engineering and Sciences
关键词：
Group ring;
Unit group;
Unitary units;
Normal complement;
Primary 16U60;
Secondary 20C05;
20E45;
D O I：
10.1007/s00013-025-02114-0
中图分类号：
学科分类号：
摘要：
Assume F is a finite field of order pf\documentclass[12pt]{minimal}
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\begin{document}$$p^f$$\end{document} and q is an odd prime for which pf-1=sqm\documentclass[12pt]{minimal}
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\begin{document}$$p^f-1=sq^m$$\end{document}, where m≥1\documentclass[12pt]{minimal}
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\begin{document}$$m \ge 1$$\end{document} and (s,q)=1\documentclass[12pt]{minimal}
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\begin{document}$$(s,q)=1$$\end{document}. In this article, we obtain the order of the symmetric and the unitary subgroup of the semisimple group algebra FCq.\documentclass[12pt]{minimal}
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\begin{document}$$FC_q.$$\end{document} Further, for the extension G of Cq=⟨b⟩\documentclass[12pt]{minimal}
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\begin{document}$$C_q = \langle b \rangle $$\end{document} by an abelian group A of order pn\documentclass[12pt]{minimal}
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\begin{document}$$p^n$$\end{document} with CA(b)={e}\documentclass[12pt]{minimal}
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\begin{document}$$C_{A}(b) = \{e\}$$\end{document}, we prove that if m>1,\documentclass[12pt]{minimal}
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\begin{document}$$m>1,$$\end{document} or (s+1)≥q\documentclass[12pt]{minimal}
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\begin{document}$$(s+1) \ge q$$\end{document} and 2n≥f(q-1)\documentclass[12pt]{minimal}
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\begin{document}$$2n \ge f(q-1)$$\end{document}, then G does not have a normal complement in V(FG).
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页码：491 / 501
页数：10
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