257 research outputs found
A new near octagon and the Suzuki tower
We construct and study a new near octagon of order which has its
full automorphism group isomorphic to the group and which
contains copies of the Hall-Janko near octagon as full subgeometries.
Using this near octagon and its substructures we give geometric constructions
of the -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 .Comment: 24 pages, revised version with added remarks and reference
Zoology of Atlas-groups: dessins d'enfants, finite geometries and quantum commutation
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
In recent work, we constructed a new near octagon from certain
involutions of the finite simple group and showed a correspondence
between the Suzuki tower of finite simple groups, , and the tower of near polygons, . 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 is similar to an
earlier characterization due to Cohen and Tits who proved that it is the unique
regular near octagon with parameters , but instead of regularity
we assume existence of an isometrically embedded dual split Cayley hexagon,
. We also give a complete classification of near hexagons of
order and use it to prove the uniqueness result for .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
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
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 and their corresponding point-line incidence geometry
. I introduce a fundamental homomorphism mapping the
(infinitely many) words of G to the permutations ruling the symmetries of
. The substructure of 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
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
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