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REPRESENTATION OF GROUPS BY AUTOMORPHISMS OF NORMAL TOPOLOGICAL SPACES
A. V. Koganov Since Research Institute of Sistem Research at RSA, Moscow. The famous fact [1, 2] of existing the exact representation for any finite Group in form the full automorphism group of finite graph was generalize in [3] . For arbitrary Group exist the exact representation in form the full automorphism group of Kolmogorov topologic space (slack type of separability T0 ). If the Group is finite that space may to chose finite, and for it may restore the finite Graph with same number of vertex which have same Automorphism Group. That Topologic Spaces and Graphs are named Topologic Image or Graph Image of Group correspond (T-image and G-image). The question about maximum type of topology separability, for which the T-image of any Group is possible, stays open. The author proof, that task is solvability on the class of normal topology (maximal type of separability T4 + T1 ). Special finite T-images exist for symmetric groups in form of discrete topologies. Minimal power of normal T-image for any other group is countable. The universal construction of that T-images exist. For event of finite Group that space have the finite subdivision with graph of connectivity, which is G-image for that Group. References 1. Koning D. Theorie der endlichen und unendlichen Graphen. Leipzig, 1936 2. Frucht R. Herstellung von Graphen mit vogegebener abstracten Grouppe. Compositio Math, v. 6 (1938), 239-250 3. A. V. Koganov. Faithful Representations of Groups by Automorphisms of Topologies. Russian Journal of Mathematical Physics, vol. 15, No 1, 2008, s. 66-76