Topological Right Gyrogroups and Gyrotransversals

Abstract

In this article we study topological right gyrogroups and gyrotransversals and discuss the situations in which it is unique. It is shown that the Einstein gyrogroup is the unique gyrotransversal to its gyroautomorphism group in its group of gyrogroup motions. Further, we show that every locally compact Hausdorff topological right gyrogroup appears as a topological gyrotransversal to a closed subgroup of a topological group in some universal sense. We also show that there are right gyrogroups (also gyrogroups) which can not be embedded as a gyrotransversal to a closed subgroup in a connected topological group to which it generates.

Key Words:

• Einstein addition
• Gyrotransversals
• Right transversals
• Right loops

2010 Mathematics Subject Classification:

• 00A69
• 00A79
• 20A05
• 20N05
• 22D05
• 83C99

ACKNOWLEDGEMENTS

This work is supported by CSIR-INDIA grant.

Notes

Communicated by A. Olshanskii.