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arXiv · 1802.06680

Extension of Maschke's theorem

Teerapong Suksumran
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Abstract

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

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Teerapong Suksumran. 2018-08-20. Extension of Maschke's theorem. https://doi.org/10.1080/00927872.2018.1530251

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