Abstract
An automorphism of a bordered compact Klein surface induces a permutation of its boundary components and we study the corresponding representations of groups of automorphisms of such surfaces in the corresponding finite symmetric groups. The principal result gives a characterization of those abstract representations of a finite group G in symmetric groups which, up to the natural equivalence, proceed from compact Klein surfaces on which G acts as a group of dianalytic automorphisms. We deal with such representations for actions of G given by a so called smooth epimorphisms Λ → G, where Λ are so called non-Euclidean crystallographic groups (NEC-groups). For such data we calculate the kernels of our representations which allow, as an application, to get some results on the minimal degree of such representations for finite abstract groups.
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The authors are very grateful to both referees for their accurate and helpful suggestions and comments concerning submitted versions of the paper which allowed us to improve it essentially.
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C. Bagiński was supported by the Grant S/WI/1/2011 of Białystok University of Technology and G. Gromadzki by the Research Grant NN 201 366436 of the Polish Ministry of Sciences and Higher Education.
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Bagiński, C., Gromadzki, G. On symmetric representations of groups of automorphism of bordered Klein surfaces. RACSAM 106, 359–369 (2012). https://doi.org/10.1007/s13398-012-0062-x
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DOI: https://doi.org/10.1007/s13398-012-0062-x
Keywords
- Klein surfaces
- Automorphisms of Klein surfaces
- Boundary of Klein surfaces
- NEC-groups
- Real forms of complex algebraic curves
- Symmetric (permutation) representation of finite groups