Narrow Results

This paper deals with an admissible AR-group, which is a group admitting an action on itself by automorphism (or, alternatively, an automorphism representation on itself) satisfying a certain nice property. Then it is shown that this group gives rise to a nonassociative group-like structure called a gyrogroup via the method of forming semidirect products. Several families of admissible AR-groups are also determined.

In this paper, we establish a strong connection between groups and gyrogroups, which provides the machinery for studying gyrogroups via group theory. Specifically, we prove that there is a correspondence between the class of gyrogroups and a class of triples with components being groups and twisted subgroups. This in particular provides a construction of a gyrogroup from a group with an automorphism of order two that satisfies the uniquely 2-divisible property. We then present various examples of such groups, including the general linear groups over $ℝ$ and $ℂ$, the Clifford group of a Clifford algebra, the Heisenberg group on a module, and the group of units in a unital C${}^{\ast }$-algebra. As a consequence, we derive polar decompositions for the groups mentioned previously.

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.