Construction of Gyrogroups of Order $2^n$ by Cyclic 2-Groups

A gyrogroup is a structure constituting from a non-empty set and a binary operation such that satisfying the left identity, and left inverse conditions, and also has the associative-like law said to be left gyroassociativity and left loop property. In this paper, we propose a method for constructing new gyrogroups of order $2^n$ by a cyclic 2-group that is ${\mathbb{Z}}_{2^n}$, where $n\ge 3$.