Read e-book online Flag-transitive Steiner Designs (Frontiers in Mathematics) PDF

By Michael Huber

ISBN-10: 3034600011

ISBN-13: 9783034600019

The monograph offers 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, staff concept, combinatorics, coding idea, and cryptography. In a sufficiently self-contained and unified demeanour the category of all flag-transitive Steiner designs is gifted. This contemporary outcome settles fascinating and hard questions which have been item of study for greater than forty years. Its evidence combines equipment from finite team idea, occurrence geometry, combinatorics, and quantity idea. The ebook incorporates a large advent to the subject, in addition to many illustrative examples. in addition, a census of a few of the main basic effects on hugely symmetric Steiner designs is given in a survey bankruptcy. The monograph is addressed to graduate scholars in arithmetic and computing device technological know-how in addition to demonstrated researchers in layout idea, finite or prevalence geometry, coding concept, cryptography, algebraic combinatorics, and extra ordinarily, discrete arithmetic.

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Extra resources for Flag-transitive Steiner Designs (Frontiers in Mathematics)

Example text

The ﬁrst results in this regard go back to J. Tits [120, Thm. 1 and 2] in 1964. He provided two beautiful characterizations of the large Mathieu-Witt designs 3-(22, 6, 1), 4-(23, 7, 1), and 5-(24, 8, 1). Let us assume that D = (X, B, I) is a Steiner t-design. Then t + 1 points of X are called independent if they are not incident with the same block. From the construction of the large Mathieu-Witt designs provided by Witt [125, 126] (cf. Chapter 4) every Steiner t-design D is isomorphic to one of the Mathieu-Witt designs 3-(22, 6, 1), 4-(23, 7, 1), and 5-(24, 8, 1) satisﬁes the following properties: (A) The full group Aut(D) of automorphisms of D acts transitively on the set of ordered subsets of D consisting of t + 1 independent points; (B) The full group Aut(D) of automorphisms of D acts transitively on the set of ordered subsets of D consisting of t + 2 points in which any t + 1 points are independent; (C) Two blocks of D which are incident with at least t − 2 common points are incident with t − 1 common points.

1 (Kantor 1985). Let D = (X, B, I) be a non-trivial Steiner 2-design, and let G ≤ Aut(D) act point 2-transitively on D. Then one of the following holds: −1 , q + 1, 1) design whose points and blocks are (1) D is isomorphic to the 2-( qq−1 the points and lines of the projective space P G(d − 1, q), and P SL(d, q) ≤ G ≤ P Γ L(d, q), or (d − 1, q) = (3, 2) and G ∼ = A7 ; d (2) D is isomorphic to a Hermitian unital UH (q) of order q, and P SU (3, q 2 ) ≤ G ≤ P Γ U (3, q2 ); (3) D is isomorphic to a Ree unital UR (q) of order q with q = 32e+1 > 3, and Re(q) ≤ G ≤ Aut(Re(q)); (4) D is isomorphic to the 2-(q d , q, 1) design whose points and blocks are the points and lines of the aﬃne space AG(d, q), and one of the following holds: (i) G ≤ AΓ L(1, q d ), (ii) G0 ☎ SL( ad , q a ), d ≥ 2a, , q a ), d ≥ 2a, (iii) G0 ☎ Sp( 2d a (iv) G0 ☎ G2 (q a ) , q even, d = 6a, (v) G0 ☎ SL(2, 3) or SL(2, 5), v = q 2 , q = 5, 7, 9, 11, 19, 23, 29 or 59, (vi) G0 ☎SL(2, 5), or G0 contains a normal extraspecial subgroup E of order 25 and G0 /E is isomorphic to a subgroup of S5 , v = 34 , (vii) G0 ∼ = SL(2, 13), v = 36 ; (5) D is isomorphic to the aﬃne nearﬁeld plane A9 of order 9, and G0 as in (4)(vi); (6) D is isomorphic to the aﬃne Hering plane A27 of order 27, and G0 as in (4)(vii); (7) D is isomorphic to one of the two Hering spaces 2-(93 , 9, 1), and G0 as in (4)(vii).

G. Higman and J. E. McLaughlin [56] proving that a ﬂagtransitive group G ≤ Aut(D) of automorphisms of a Steiner 2-design D is necessarily primitive on the points of D. They posed the problem of classifying all ﬁnite ﬂag-transitive projective planes, and showed that such planes are Desarguesian if their orders are suitably restricted. Much later W. M. Kantor [80] determined all such planes apart from the still open case when the group of automorphisms is a Frobenius group of prime degree. His proof involves detailed knowledge of primitive permutation groups of odd degree based on the classiﬁcation of the ﬁnite simple groups.

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