Involutive groups, unique 2-divisibility, and related gyrogroup structures

Abstract

© 2017 World Scientific Publishing Company. 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 and the Clifford group of a Clifford algebra, the Heisenberg group on a module, and the group of units in a unital C∗-algebra. As a consequence, we derive polar decompositions for the groups mentioned previously.