关于扭群环的半简单性和广义maschke定理

IF 0.5 Q3 MATHEMATICS

Asian-European Journal of Mathematics Pub Date : 2023-10-21 DOI:10.1142/s1793557123502200

Alanka Thomas, P.G. Romeo

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引用次数: 0

摘要

经典Maschke定理给出了群代数在域上具有半简单性的一个条件。由于群代数是扭群环的一种特殊情况，我们将Maschke定理推广到除环上的扭群环上，证明了如果一个除环[公式:见文]的特征不能除有限群[公式:见文]的阶，则扭群环[公式:见文]是半单的。然后，我们证明了由[公式:见文]的[公式:见文]的分裂扩展引起的扭群环的逆定理。由于在除法环上，扭转群环与群格之间存在一一对应关系，我们得到了扭转群环在群格上的半简单性的结论。

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On the semisimplicity of twisted group rings and generalized maschke's theorem

Classical Maschke’s theorem gives a condition for the semisimplicity of group algebras over a field. As group algebras are a particular case of twisted group rings, we extend Maschke’s theorem to twisted group rings over division rings by proving that if the characteristic of a division ring [Formula: see text] does not divide the order of a finite group [Formula: see text], twisted group ring [Formula: see text] is semisimple. Then we prove the converse of this theorem for twisted group rings arising from split extensions of [Formula: see text] by [Formula: see text]. As there is a one-to-one correspondence between twisted group rings and group lattices over division rings, we conclude with the consequence of the semisimplicity of twisted group rings on group lattices.

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来源期刊

Asian-European Journal of Mathematics MATHEMATICS-

CiteScore

1.30

自引率

12.50%

发文量

169

期刊介绍： Asian-European Journal of Mathematics is an international journal which is devoted to original research in the field of pure and applied mathematics. The aim of the journal is to provide a medium by which new ideas can be discussed among researchers from diverse fields in mathematics. It publishes high quality research papers in the fields of contemporary pure and applied mathematics with a broad range of topics including algebra, analysis, topology, geometry, functional analysis, number theory, differential equations, operational research, combinatorics, theoretical statistics and probability, theoretical computer science and logic. Although the journal focuses on the original research articles, it also welcomes survey articles and short notes. All papers will be peer-reviewed within approximately four months.