Involutory decomposition of groups into twisted subgroups and subgroups. (English) Zbl 0944.20053

Authors’ abstract: An involutory decomposition is a decomposition, due to an involution, of a group into a twisted subgroup and a subgroup. We study unexpected links between twisted subgroups and gyrogroups. Twisted subgroups arise in the study of problems in computational complexity. In contrast, gyrogroups are group-like structures which first arose in the study of Einstein’s velocity addition in the special theory of relativity. In particular, we show that every gyrogroup is a twisted subgroup and that, under general specified conditions, twisted subgroups are gyrocommutative gyrogroups. Moreover, we show that gyrogroups abound in group theory and that they possess rich structure.


20N05 Loops, quasigroups
20E22 Extensions, wreath products, and other compositions of groups
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