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Gyrogroups and the decomposition of groups into twisted subgroups and subgroups. (English) Zbl 1066.20068

The following propositions are proved: any left gyrogroup is a gyrotransversal of some group; any group \(K(\cdot)\) can be turned into a left gyrogroup with operation \(a\bullet b=ab^a\); the associated left gyrogroup \(K(\bullet)\) is a group iff the group \(K(\cdot)\) is nilpotent of class 2; the associated left gyrogroup \(K(\bullet)\) is a gyrogroup iff the group \(K(\cdot)\) is central by a 2-Engel group. An example of a non-group and non-gyrocommutative matrix gyrogroup is constructed.

MSC:

20N05 Loops, quasigroups

Software:

Magma
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