The Isomorphism Problem of Normalized Unit Groups of Group Algebras of a Class of Finite 2-groups

IF 0.8 3区 数学 Q2 MATHEMATICS
Yu Lei Wang, He Guo Liu
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引用次数: 0

Abstract

Let p be a prime and \({\mathbb{F}_p}\) be a finite field of p elements. Let \({\mathbb{F}_p}G\) denote the group algebra of the finite p-group G over the field \({\mathbb{F}_p}\) and \(V({\mathbb{F}_p}G)\) denote the group of normalized units in \({\mathbb{F}_p}G\). Suppose that G and H are finite p-groups given by a central extension of the form

$$1 \to {\mathbb{Z}_{{p^m}}} \to G \to {\mathbb{Z}_p} \times \cdots \times {\mathbb{Z}_p} \to 1$$

and \({G^\prime } \cong {\mathbb{Z}_p},\,\,m \ge 1\). Then \(V({\mathbb{F}_p}G) \cong V({\mathbb{F}_p}H)\) if and only if GH. Balogh and Bovdi only solved the isomorphism problem when p is odd. In this paper, the case p = 2 is determined.

一类有限2群群代数的归一化单位群的同构问题
设p是素数,\({\mathbb{F}_p}\)是p元素的有限域。设\({\mathbb{F}_p}G\)表示域\({\mathbb{F}_p}\)上有限p群G的群代数,\(V({\mathbb{F}_p}G)\)表示\({\mathbb{F}_p}G\)中的归一化单位群。假设G和H是有限的p群,由形式$$1 \to {\mathbb{Z}_{{p^m}}} \to G \to {\mathbb{Z}_p} \times \cdots \times {\mathbb{Z}_p} \to 1$$和\({G^\prime } \cong {\mathbb{Z}_p},\,\,m \ge 1\)的中心扩展给出。则\(V({\mathbb{F}_p}G) \cong V({\mathbb{F}_p}H)\)当且仅当G = h, Balogh和Bovdi仅在p为奇数时解决了同态问题。本文确定了p = 2的情况。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
CiteScore
1.00
自引率
0.00%
发文量
138
审稿时长
14.5 months
期刊介绍: Acta Mathematica Sinica, established by the Chinese Mathematical Society in 1936, is the first and the best mathematical journal in China. In 1985, Acta Mathematica Sinica is divided into English Series and Chinese Series. The English Series is a monthly journal, publishing significant research papers from all branches of pure and applied mathematics. It provides authoritative reviews of current developments in mathematical research. Contributions are invited from researchers from all over the world.
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