Strong near subgroups and left gyrogroups
type: Item
instance of: scholarly article
title: Strong near subgroups and left gyrogroups
zbMATH Open document ID: 1019.20031
DOI: 10.1016/S0021-8693(02)00536-7
author: Tomás Feder
published in: Journal of Algebra
publication date: 2003-04-02T00:00:00Z
review text: Strong near subgroups of finite groups are defined as twisted subgroups for odd order groups enjoying some additional properties. The author generalizes Glauberman's result and shows that if a twisted subgroup \(K\) is order-dividing in \(G\), then \(K\) is a strong near subgroup of \(G\) and conversely, if \(K\) is a strong near subgroup of \(G\), then \(K\) is order-dividing in \(G\). Gyrogroups arose in the study of the Thomas precession in the special theory of relativity. \textit{T. Foguel} and \textit{A. A. Ungar} [J. Group Theory 3, No.~1, 27-46 (2000; Zbl 0944.20053)] showed that every twisted subgroup of an odd order group can be viewed as a gyrocommutative gyrogroup. (A gyrocommutative gyrogroup is the same as a Bruck loop or \(K\)-loop, a left gyrogroup is the same as a left loop having the left inverse property and having the \(A_l\) property.) In this paper there are studied the relations among a near subgroup, strong near subgroup, left gyrogroup and gyrocommutative gyrogroup. The exhibited examples illustrate some situations.
Mathematics Subject Classification ID: 20N05
Mathematics Subject Classification ID: 20D60
zbMATH DE Number: 1891317
zbMATH Keywords: twisted subgroups
zbMATH Keywords: odd order groups
zbMATH Keywords: gyrocommutative gyrogroups
zbMATH Keywords: Bruck loops
zbMATH Keywords: strong near subgroups
zbMATH Keywords: left loops
reviewed by: Elena Broží ková
MaRDI profile type: Publication
cites work: Q3708174
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description: scientific article; zbMATH DE number 1891317