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The genus of a group Medalen, David Norman


This article is a survey of the two definitions of the genus of a finite group that are prominent in the mathematical literature. We call them the Cayley genus and the Burnside genus of a group, but we do not intend to suggest that either mathematician was responsible for the original definitions. Each definition gives a scheme for classifying finite groups by topological criteria. The Cayley genus is determined by embedding a certain graph for the group in a surface of least possible genus. The genus of this surface is then defined as the Cayley genus of the group. The Burnside genus of a group is determined by showing that the group is isomorphic to a group of conformal homeomorphisms of a surface of finite genus, and is not isomorphic to such a group for any surface of lesser genus. The genus of the surface is defined to be the Burnside genus of the group. This paper tries to present a broad mathematical background for each definition of group genus. We attempt to show most of the ideas and methods employed in the research, and most of the significant known theory and results. Proofs are sporadically included, but only to convey a sense of the nature of the research.

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