Representation of groups by automorphisms of normal topological spaces

Koganov A.V.
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The famous fact [3, 5] of existence of an exact representation for any finite group in the form of the full automorphism group of a finite graph was generalize in [4]. For an arbitrary group exact representation exists in the form of the full automorphism group of Kolmogorov topological space (weak type of separability T0). For a finite group a finite space may be chosen, thus allowing to restore a finite graph with the same number of vertices and having the same automorphism group. Such topological spaces and graphs are called topological imprints and graph imprints of a group (T-imprints and G-imprints, respectively). The question of maximum type of separability of a topological space for which T-imprint can be obtained for any group is open. The author proves that the problem can be solved for the class of normal topology (maximal type of separability T4+T0). Special finite T-imprint for a symmetric group may be obtained as a discrete topology; for any other group minimal cardinality of normal T-imprint is countable. There is a generic procedure to construct a T-imprint for any group. For a finite group this procedure allows finite space partitioning into subspaces having G-imprint of the original group as their connectivity graphs.

Keywords: group theory, automorphisms, topological spaces
Citation in English: Koganov A.V. Representation of groups by automorphisms of normal topological spaces // Computer Research and Modeling, 2009, vol. 1, no. 3, pp. 243-249
Citation in English: Koganov A.V. Representation of groups by automorphisms of normal topological spaces // Computer Research and Modeling, 2009, vol. 1, no. 3, pp. 243-249
DOI: 10.20537/2076-7633-2009-1-3-243-249
URL: http://crm-en.ics.org.ru/journal/article/1585/
Creative Commons License This work is licensed under a Creative Commons Attribution-NoDerivs 3.0 Unported License.
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