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Flag transitive Steiner Systems after Michael Huber

Flag transitive Steiner Systems after Michael Huber. Francis Buekenhout Université libre de Bruxelles Académie Royale de Belgique Classe des Sciences Lecture for Finite Geometry Irsee 13-09-06. Michael Huber (1972- Universität Tübingen. Steiner System. S=S(t,k,v)

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Flag transitive Steiner Systems after Michael Huber

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  1. Flag transitive Steiner Systemsafter Michael Huber Francis Buekenhout Université libre de Bruxelles Académie Royale de Belgique Classe des Sciences Lecture for Finite Geometry Irsee 13-09-06

  2. Michael Huber(1972-Universität Tübingen

  3. Steiner System • S=S(t,k,v) • Finite set of Points equipped with subsets called Blocks • There are v points. • Each block has k points • Every set of t points in a unique block

  4. Derived Steiner System • S=S(t,k,v) • Fix point p • Then Sp, consting of all points other than p and all blocks on p with p deleted, is a • S(t-1, k-1, v-1)

  5. Huber’s LemmaAut S is 2-transitive on points • S flag transitive, p a point • In Sp, (Aut S)p is transitive on the blocks • By Block’s Lemma it is transitive on the points • So, Aut S is 2-transitive on the points • FLAG TRANSITIVE implies 2-TRANSITIVE

  6. Non-trivial Steiner System • We assume t≥3 but t= 1 or 2 is not bad • Forget t=k • Forget k=v • So 2<t<k<v

  7. Flag transitive S • Flag of S: pair (p, B) where p is a point and B is a block containing p • Aut S: group of automorphisms of S • S flag transitive: Aut S acts transitively on the flags of S

  8. Block’s Lemma 1965 • R.E. Block, Transitive groups of collineations on certain designs, Pacific J. Math. 15, 1965, 13-18 • If G is a group of automorphisms of a rank 2 incidence structure and the number of points is smaller or equal to the number of blocks. • Then the number of point orbits of G is not larger than the number of line orbits of G • Consequence: if G is block transitive then G is point transitive

  9. Huber’s Deep Theorem • IF S=S(t,k,v) flag transitive, non-trivial, 3≤ t THEN S is known Observe: this remains true if you remove « non-trivial ». It remains also true for t=2 (by BDDKLS) except for an open case.

  10. About the proof 1First came Cameron-Praeger • For t≥ 7, the result is due to Cameron-Praeger (In Deinze 1992) • P.J. Cameron and C. E. Praeger, Block -transitive t-designs, II, large t. In Finite Geometry and Combinatorics (Deinze 1992), Editors: F. De Clerck, e.a. London Math. Soc. Lecture Note Series 191, Cambridge U.P., 1993, 103-119.

  11. About the proof 2Apply Huber’s Lemma • This is leaving four cases namely t=3, t=4, t=5, t=6. • By Huber’s Lemma, Aut S is 2-transitive on the points of S

  12. About the Proof 3Apply classification of 2-transitive groups • The 2-transitive permutation groups are known. This is due to combined work of • Curtis, Kantor, Seitz ( 1976) • Gorenstein (1982) • Hering (1974, 1985) • Huppert (1957) • Kantor (1985) • Maillet (1895)

  13. About the Proof 4Apply classification of 2-transitive groups • The 2-transitive permutation groups are known. • They appear in a list of 21 classes or types. • Task of Huber now: deal with 84 cases. • In each case he knows: t, the number v of points, the automorphism group G and its action on S, in particular the structure of a point stabilizer. The size k of a block and the nature of a block remain unknown. Huber’s miracle: the difficulty can be overcome.

  14. A great ancestor: Heinz Lüneburg 1965 • The work of Lüneburg has been a truly inspiring source for Huber. • H. Lüneburg, Fahnenhomogenen Quadrupelsysteme, Math. Zeit. 89, 1965, 82-90 • M. Huber, Classification of flag-transitive Steiner Quadruple Systems, JCT(A), 94, 2001, 180-190

  15. A great ancestor: Jacques Tits (Roma 1963) • J. Tits, Sur les systèmes de Steiner associés aux trois « grands » groupes de Mathieu, Rendiconti di Mat. 23, 1964,166-184 • This remained unknown to Huber until recently • Tits uses a concept of t+1 independent points namely t+1 points not on a block (that I would call an apartment)

  16. A great ancestor: Jacques Tits (Roma 1963) 2 • Theorem 1 (Tits) IF S is transitive on ordered apartments and any two blocks intersecting in t-2 points at least do intersect in t-1 points, THEN S is known. • Huber generalizes this result. • Theorem 2 of Tits is also generalized by Huber.

  17. Huber’s list 1 (1) Affine space. Here, t=3, v=2d, k=4, points and planes of AG(d, 2) and one of (1.1) d≥3 and G= AGL(d,2) (1.2) d=3 and G=AGL(1,8) or AG*L(1,8) (1. 3) d=4 and G0=A7 (1.4) d=5 and G=AG*L(1,32) G* means gamma

  18. Huber’s list 2 (2) Projective line. Here, t=3, v=qe+1, k=q+1, q is a prime power, q≥3, e is an integer, e≥2. Points are those from the projective line over GF(qe). Blocks are the sublines over GF(q). Also, PSL(2,q) ≤ G≤ PG*L(2,q) and it is allowed that G is PSL(2,q) for e odd.

  19. Huber’s list 3 (3) Extended Netto System. Here t=3, v=q+1, k=4, the points are those of a projective line over GF(q) where q is a prime power with q=7(mod 12), the blocks are the transforms under PSL(2,q) of {0, 1, e, infinity}where e is a primitive sixth root of unity in GF(q). Also PSL(2,q) ≤ G ≤ PS*L(2,q)

  20. Huber’s list 4 (4) Witt-Mathieu system. Here, one of the following holds: (4.1) t=3, v=22, k=6 G=M22 or M22.2 (4.2) t=4, v=11, k=5 G=M11 (4.3) t=4, v=23, k=7 G=M23 (4.4) t=5, v=12, k=6 G=M12 (4.5) t=5, v=24, k=8 G=M24 or PSL(2, 23)

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