Extension of Maschke’s theorem

Abstract

In the present article, we examine linear representations of finite gyrogroups, following their group-counterparts. In particular, we prove Maschke’s theorem for gyrogroups, along with its converse. This suggests studying the left regular action of a gyrogroup $(G,\oplus )$ on the function space $$L^{\text{gyr}}(G)=\{f\in L(G):\forall a,x,y,z\in G,f(a\oplus \text{gyr}[x,y]z)=f(a\oplus z)\}$$ in a natural way, where L(G) is the space of all functions from G to a field.

Keywords:

• Gyrogroup
• gyrogroup action
• left regular representation
• linear representation
• Maschke’s theorem

2010 MATHEMATICS Subject Classification:

• Primary 20G05
• Secondary 20N05

Acknowledgments

The author would like to thank Keng Wiboonton for his collaboration. He is grateful to the referee for careful reading of the manuscript and invaluable comments. This research is part of the project Gyrogroups and their linear representations, funded by the Institute for Promotion of Teaching Science and Technology (IPST), Thailand.

Additional information

Funding

This research was supported by the Research Fund for DPST Graduate with First Placement, Year 2016, under grant No. 028/2559 and Chiang Mai University.