On commutator-inversion invariant groups and gyrogroup construction

Abstract

This paper introduces the notion of commutator-inversion invariance and studies several algebraic properties of commutator-inversion invariant groups. Then, a characterization of 2-Engel groups is given, and it is shown that any group whose central quotient is commutator-inversion invariant gives rise to a non-associative structure called a gyrogroup. This method yields three non-degenerate gyrogroups of order 16 as concrete examples.