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Bull. Korean Math. Soc. 2021; 58(1): 225-234

Online first article September 3, 2020      Printed January 31, 2021

https://doi.org/10.4134/BKMS.b200212

Copyright © The Korean Mathematical Society.

Subpermutable subgroups of skew linear groups and unit groups of real group algebras

Le Qui Danh, Nguyen Trung Nghia, Nguyen Kim Ngoc

University of Architecture Ho Chi Minh City; Vietnam National University; Vietnam National University

Abstract

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Let $D$ be a division ring and $n>1$ be an integer. In this paper, it is shown that if $D\ne \mathbb{F}_3$, then every subpermutable subgroup of the general skew linear group ${\rm GL}_n(D)$ is normal. By applying this result, we show that every subpermutable subgroup of the unit group $(\mathbb{R} G)^*$ of the real group algebras $\mathbb{R} G$ of finite groups $G$ is normal in $(\mathbb{R} G)^*$.

Keywords: Permutable subgroup, quasinormal subgroup, subpermutable subgroup, general linear group, real group algebra

MSC numbers: 16K20, 20B99

Supported by: The third author was funded by University of Science, VNU-HCM, under grant number T2019-04

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