By Michael Huber
ISBN-10: 3034600011
ISBN-13: 9783034600019
The monograph presents the 1st complete dialogue of flag-transitive Steiner designs. this can be a valuable a part of the learn of hugely symmetric combinatorial configurations on the interface of numerous mathematical disciplines, like finite or occurrence geometry, workforce conception, combinatorics, coding idea, and cryptography. In a sufficiently self-contained and unified demeanour the type of all flag-transitive Steiner designs is gifted. This fresh end result settles attention-grabbing and not easy questions which were item of study for greater than forty years. Its evidence combines equipment from finite team conception, occurrence geometry, combinatorics, and quantity theory.
The ebook encompasses a wide advent to the subject, besides many illustrative examples. additionally, a census of a few of the main normal effects on hugely symmetric Steiner designs is given in a survey chapter.
The monograph is addressed to graduate scholars in arithmetic and desktop technological know-how in addition to tested researchers in layout idea, finite or prevalence geometry, coding concept, cryptography, algebraic combinatorics, and extra often, discrete mathematics.
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Extra resources for Flag-transitive Steiner Designs
Example text
Buekenhout, A. Delandtsheer, J. Doyen, P. B. Kleidman, M. W. Liebeck, and J. , flag-transitive Steiner 2-designs. Their result, which also relies on the classification of the finite simple groups, starts with the result of Higman and McLaughlin and uses the O’Nan-Scott Theorem for finite primitive permutation groups. For the incomplete case with a 1-dimensional affine group of automorphisms, we refer to [23, Sect. 4], [81, Sect. 3], and [21]. 3 (Buekenhout et al. 1990). Let D = (X, B, I) be a Steiner 2-design, and let G ≤ Aut(D) act flag-transitively on D.
Point 0. Hence GB ≤ G0 , and 0 ∈ As G is 2-transitive on points, we have |G| = v(v − 1)a with a | d. 15 yields v − 2 = (k − 1)(k − 2) a if x ∈ B. 3) As GB fixes some y ∈ / B, it follows that |GxB | |Gxy | = a. If G0x fixes three or more distinct points, then G0x would fix some block B ∈ B. Thus, we have a |GxB |, and therefore v − 2 = (k − 1)(k − 2). 16 (b) that v − 2 > (k − 1)(k − 2), a contradiction. Hence, G0x fixes only 0 and x. Then G0x must contain a field automorphism of order d, and we conclude that G = AΓ L(1, 2d ).
17, we have k ≤ pa + 1, a contradiction. Therefore, B is contained completely in e1 . Hence, as G is flag-transitive, we may conclude that each block lies in an affine line. But, by the definition of Steiner 3-designs, any three distinct non-collinear points must also be incident with a unique block, a contradiction. 3. Groups of Automorphisms of Affine Type 51 For d ≥ 3a, we consider ( ad × ad )-matrices of the form ⎛ ⎞ 1 0 0 ··· 0 ⎜x1 ⎟ ⎜ ⎟ ⎜0 ⎟ ⎜ ⎟ B i Ai = ⎜ ⎟ , 1 ≤ i ≤ ad − 1, x1 ∈ GF (pa ) arbitrary, ⎜ ..
Flag-transitive Steiner Designs by Michael Huber
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