## 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

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