Two theorems on direct products of gyrogroups

Abstract

We extend two standard theorems on groups to gyrogroups: the direct product theorem and the cancellation theorem for direct products. Firstly, we prove that under a certain condition a gyrogroup G can be decomposed as the direct product of two subgyrogroups. Secondly, we prove that finite gyrogroups can be cancelled in direct products: if A≅B, then A×H≅B×K or H×A≅K×B implies H≅K, where A,B,H, and K are finite gyrogroups.

Keywords

• direct product theorem
• cancellation theorem
• decomposition of gyrogroup
• gyrogroup
• homomorphism