Geometry of Sporadic Groups: Volume 1, Petersen and Tilde Geometries
This book is the first volume in a two-volume set, which will provide the complete proof of classification of two important classes of geometries, closely related to each other: Petersen and tilde geometries. There is an infinite family of tilde geometries associated with nonsplit extensions of symplectic groups over a field of two elements. Besides that there are twelve exceptional Petersen and tilde geometries. These exceptional geometries are related to sporadic simple groups, including the famous Monster group and this volume gives a construction for each of the Petersen and tilde geometries that provides an independent existence proof for the corresponding automorphism group. Important applications of Petersen and tilde geometries are considered, including the so-called Y-presentations for the Monster and related groups, and a complete identification of Y-groups is given. This is an essential purchase for researchers in finite group theory, finite geometries and algebraic combinatorics.
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2-cover 3-element subset action of G acts transitively adjacent amalgam of maximal bijection Borel subgroup collinearity graph complement conjugate contains corresponding cosets Coxeter denote distance-transitive easy edge elementary abelian elements of type flag-transitive automorphism group full automorphism group G acts geometrical subgraph Gi(x girth Golay code group G Hence homomorphism hyperoval images implies incidence relation incidence system intersection involution kernel Leech graph Leech lattice Lemma Let G locally projective graph maps isomorphically Mathieu groups Matu maximal flag Monster group natural module non-trivial normal octads orthogonal pairs parabolic geometry parabolic subgroups permutes Petersen graph plane of order preimage projective plane Proof residue respectively result follows Section semidirect product setwise stabilizer sextet simply connected Steiner system subamalgam subgeometry subgroups of order suborbit diagram subspaces Sylow 2-subgroup Sym3 T-geometry tilde geometry triangle trios uniquely determined universal completion valency vectors vertex-transitive vertices