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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.

MSC:

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