Strong near subgroups and left gyrogroups
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.
scientific article; zbMATH DE number 1891317