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

• Main Author: Suksumran T.
• Format: Journal
• Published: 2017
• Online Access: https://www.scopus.com/inward/record.uri?partnerID=HzOxMe3b&scp=84979256238&origin=inward; http://cmuir.cmu.ac.th/jspui/handle/6653943832/40383
• Institution: Chiang Mai University

© 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.