Geometry of Sporadic Groups II: Representations and Amalgams by A. A. Ivanov, S. V. Shpectorov

By A. A. Ivanov, S. V. Shpectorov

This moment quantity in a two-volume set presents an entire self-contained facts of the class of geometries linked to sporadic easy teams: Petersen and tilde geometries. It encompasses a learn of the representations of the geometries into account in GF(2)-vector areas in addition to in a few non-Abelian teams. The imperative half is the category of the amalgam of maximal parabolics, linked to a flag transitive motion on a Petersen or tilde geometry. in terms of their systematic therapy of crew amalgams, the authors identify a deep and significant mathematical consequence.

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Extra info for Geometry of Sporadic Groups II: Representations and Amalgams (Encyclopedia of Mathematics and its Applications 91)

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Hence the images in E/Y(Q) of B, and B ; coincide. Since {i,j} was an arbitrary edge of £ and the latter is connected, we obtain (i). 5) we observed that E(x) < Y whenever x $ fty. 5). • A geometric hyperplane ft in y is said to be acceptable if 7(ft) f E. 6) every fty is acceptable. Thus the number of represented subgroups in E (the cardinality of S) is at most the number of acceptable hyperplanes in Sf. Now in order to bound the dimension of Vc it is sufficient to bound the dimension of VY for a represented subgroup Y in E.

Let e be a preimage of e in 0 . Then a vertex / e 0 can be identified with the unique element in Q which maps 'e onto / and under this identification 8 is a homomorphism of Q onto Q. Since 8 is a covering of graphs, it induces a bijection /? ). We claim that (y) are adjacent in 0 , which means that the element a := q>(x)q>{y) belongs to the set (p(Tl) of generators.

Define a mapping \p from Va into the direct product of Z(H[) x ... ,am(u)), where a,-(c) = 0 if v € Vt and a,(u) = 1 otherwise. It is easy to see that xp is a representation homomorphism of (Va, ) is a representation of Sf such that V is generated by the images under q> of the points from a geometric hyperplane H in Sf. Then the product (V, q>) x (Z(H), q>u) possesses a proper representation • homomorphism onto (V, cp), in particular the latter is not universal.

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