257 research outputs found

    A new near octagon and the Suzuki tower

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    We construct and study a new near octagon of order (2,10)(2,10) which has its full automorphism group isomorphic to the group G2(4):2\mathrm{G}_2(4){:}2 and which contains 416416 copies of the Hall-Janko near octagon as full subgeometries. Using this near octagon and its substructures we give geometric constructions of the G2(4)\mathrm{G}_2(4)-graph and the Suzuki graph, both of which are strongly regular graphs contained in the Suzuki tower. As a subgeometry of this octagon we have discovered another new near octagon, whose order is (2,4)(2,4).Comment: 24 pages, revised version with added remarks and reference

    Zoology of Atlas-groups: dessins d'enfants, finite geometries and quantum commutation

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    Every finite simple group P can be generated by two of its elements. Pairs of generators for P are available in the Atlas of finite group representations as (not neccessarily minimal) permutation representations P. It is unusual but significant to recognize that a P is a Grothendieck's dessin d'enfant D and that most standard graphs and finite geometries G-such as near polygons and their generalizations-are stabilized by a D. In our paper, tripods P -- D -- G of rank larger than two, corresponding to simple groups, are organized into classes, e.g. symplectic, unitary, sporadic, etc (as in the Atlas). An exhaustive search and characterization of non-trivial point-line configurations defined from small index representations of simple groups is performed, with the goal to recognize their quantum physical significance. All the defined geometries G' s have a contextuality parameter close to its maximal value 1.Comment: 19 page

    Characterizations of the Suzuki tower near polygons

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    In recent work, we constructed a new near octagon G\mathcal{G} from certain involutions of the finite simple group G2(4)G_2(4) and showed a correspondence between the Suzuki tower of finite simple groups, L3(2)<U3(3)<J2<G2(4)<SuzL_3(2) < U_3(3) < J_2 < G_2(4) < Suz, and the tower of near polygons, H(2,1)H(2)DHJG\mathrm{H}(2,1) \subset \mathrm{H}(2)^D \subset \mathsf{HJ} \subset \mathcal{G}. Here we characterize each of these near polygons (except for the first one) as the unique near polygon of the given order and diameter containing an isometrically embedded copy of the previous near polygon of the tower. In particular, our characterization of the Hall-Janko near octagon HJ\mathsf{HJ} is similar to an earlier characterization due to Cohen and Tits who proved that it is the unique regular near octagon with parameters (2,4;0,3)(2, 4; 0, 3), but instead of regularity we assume existence of an isometrically embedded dual split Cayley hexagon, H(2)D\mathrm{H}(2)^D. We also give a complete classification of near hexagons of order (2,2)(2, 2) and use it to prove the uniqueness result for H(2)D\mathrm{H}(2)^D.Comment: 20 pages; some revisions based on referee reports; added more references; added remarks 1.4 and 1.5; corrected typos; improved the overall expositio

    Products of conjugacy classes in finite and algebraic simple groups

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    We prove the Arad-Herzog conjecture for various families of finite simple groups- if A and B are nontrivial conjugacy classes, then AB is not a conjugacy class. We also prove that if G is a finite simple group of Lie type and A and B are nontrivial conjugacy classes, either both semisimple or both unipotent, then AB is not a conjugacy class. We also prove a strong version of the Arad-Herzog conjecture for simple algebraic groups and in particular show that almost always the product of two conjugacy classes in a simple algebraic group consists of infinitely many conjugacy classes. As a consequence we obtain a complete classification of pairs of centralizers in a simple algebraic group which have dense product. In particular, there are no dense double cosets of the centralizer of a noncentral element. This result has been used by Prasad in considering Tits systems for psuedoreductive groups. Our final result is a generalization of the Baer-Suzuki theorem for p-elements with p a prime at least 5.Comment: 36 page

    Two-letter words and a fundamental homomorphism ruling geometric contextuality

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    It has recently been recognized by the author that the quantum contextuality paradigm may be formulated in terms of the properties of some subgroups of the two-letter free group GG and their corresponding point-line incidence geometry G\mathcal{G}. I introduce a fundamental homomorphism ff mapping the (infinitely many) words of G to the permutations ruling the symmetries of G\mathcal{G}. The substructure of ff is revealing the essence of geometric contextuality in a straightforward way.Comment: 18 pages, 11 figures, 2 tables to appear in "Symmetry: Culture and Science

    Maximal subgroups of sporadic groups

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    A systematic study of maximal subgroups of the sporadic simple groups began in the 1960s. The work is now almost complete, only a few cases in the Monster remaining outstanding. We give a survey of results obtained, and methods used, over the past 50 years, for the classification of maximal subgroups of sporadic simple groups, and their automorphism groups.Comment: To appear in Proceedings of the Princeton Conference, November 2015, AMS Contemporary Mathematics series. Version 2 with a minor correction to the last tabl
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