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Title: A characterization of the Petersen-type geometry of the McLaughlin group
Authors: Baumeister, Barbara
Ivanov, A. A.
Pasechnik, Dmitrii V.
Keywords: DRNTU::Science::Mathematics::Applied mathematics
Issue Date: 2000
Source: Baumeister, B., Ivanov, A. A., & Pasechnik, D. V. (2000). A characterization of the Petersen-type geometry of the McLaughlin group. Mathematical proceedings of the Cambridge philosophical society, 128(1), 21-44.
Series/Report no.: Mathematical proceedings of the Cambridge philosophical society
Abstract: The McLaughlin sporadic simple group McL is the flag-transitive automorphism group of a Petersen-type geometry g=g(McL) with the diagram where the edge in the middle indicates the geometry of vertices and edges of the Petersen graph. The elements corresponding to the nodes from the left to the right on the diagram P_3^3 are called points, lines, triangles and planes, respectively. The residue in g of a point is the P^3-geometry g(Mat22) of the Mathieu group of degree 22 and the residue of a plane is the P^3-geometry g(Alt7) of the alternating group of degree 7. The geometries g(Mat22) and g(Alt7) possess 3-fold covers g(3Mat22) and g(3Alt7) which are known to be universal. In this paper we show that g is simply connected and construct a geometry g ̃ which possesses a 2-covering onto g. The automorphism group of g ̃ is of the form 323McL; the residues of a point and a plane are isomorphic to g(3Mat22) and g(3Alt7), respectively. Moreover, we reduce the classification problem of all flag-transitive P_n^m-geometries with n, m ≥ 3 to the calculation of the universal cover of g ̃.
Rights: © 2000 Cambridge University Press. This paper was published in Mathematical Proceedings of the Cambridge Philosophical Society and is made available as an electronic reprint (preprint) with permission of Cambridge University Press. The paper can be found at the publisher’s official URL: []. One print or electronic copy may be made for personal use only. Systematic or multiple reproduction, distribution to multiple locations via electronic or other means, duplication of any material in this paper for a fee or for commercial purposes, or modification of the content of the paper is prohibited and is subject to penalties under law.
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