On Dihedralized Gyrogroups and Their Cayley Graphs

Abstract

The method of constructing the generalized dihedral group as a semidirect product of an abelian group and the group Z2 of integers modulo 2 is extended to the case of gyrogroups. This leads to the study of a new class of gyrogroups, which includes generalized dihedral groups and dihedral groups as a special case. In this article, we show that any dihedralizable gyrogroup can be enlarged to a dihedralized gyrogroup. Then, we establish algebraic properties of dihedralized gyrogroups as well as combinatorial properties of their Cayley graphs.