flag-transitive linear spaces.

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1 Linear spaces exactly Characterisation one line incident with any One-dimensional two given points. examples Finite linear spaces can Pauley-nomials be used to make Open problem experimental designs and error correcting codes. The collineations of a linear space are the bijections from the linear space to itself which preserve incidence. A linear space is called flag-transitive if for any pair of points P, Q and Projective any lines l through P and permutation m through Q there is a collineation polynomials mapping P to Q and l to m. Flag-transitive linear spaces have a very high level of symmetry. flag-transitive linear spaces. Using the classification of finite simple groups, Buekenhout, Delandtsheer, Doyen, Kleidman, Liebeck and Saxl have essentially given a classification of the finite flag-transitive linear spaces. The only remaining case is when the transitive group is a subgroup of the one dimensional affine group AΓL1(q). This is called the one dimensional affine case. Kantor has given John Bamberg some classes of one dimensional affine linear spaces, but there are many more which do not fit into any of these classes. A line Centre spreadfor is athe set of Mathematics lines which partition of Symmetry the projective and space Computation, PGd(q). A line spread which admits a transitivethe groupuniversity can be used of to construct Western a flag-transitive Australia linear space. Using the GAP computer algebra system, we will search for and classify transitive line spreads of some projective spaces and we will construct an infinite class of transitive line spreads which give new one dimensional affine linear spaces. July 13, 2011 On joint work with Michael Pauley.

2 Linear spaces Finite linear space A finite linear space L is a geometry of finitely many points and lines such that every two points lie on a unique line. We say that L is regular if every line has a constant k number of points. 2 (v, k, 1) design

3 Flag Incident point and line pair (P, l). Automorphisms An automorphism φ of L is a permutation of the points which maps lines to lines (thought of as sets of points) and preserves incidence P l P φ l φ.

4 Theorem (D. G. Higman and McLaughlin 1961) If G acts transitively on the flags of a finite linear space L, then G acts primitively on the points of L. Theorem (Buekenhout, Delandtsheer, Doyen 1988)... furthermore, G is affine or almost simple. O Nan-Scott analysis Title Brief description Affine T n G AGL(n, p), G 0 irreducible Holomorph of a simple group T T G Hol(T ) Holomorph of a compound group T n T n G Hol(T n ) Almost simple T G Aut(T ) Twisted wreath T n G Hol(T n ) Simple diagonal T n G T n.(out(t ) S n) Compound diagonal T kl G (Simple diag.) l S k Product action T n G (Almost simple) S n

5 Theorem (W. M. Kantor 1985) Suppose that G is a 2-transitive group of automorphisms of a nontrivial finite linear space L. Then L is one of the following: G almost simple G affine Desarguesian projective space Hermitian unital Ree unital Desarguesian affine space Hering plane of order 27 Nearfield plane of order 9 Two Hering designs with v = 3 6 and k = 3 2

6 Theorem (Buekenhout, Delandtsheer, Doyen, Kleidman, Liebeck, Saxl 1990) Suppose that G is a flag-transitive group of automorphisms of a nontrivial finite linear space L. Then either: L is one of the following examples: G almost simple G affine or one-dimensional affine. Desarguesian projective spaces Hermitian unitals Ree unitals Witt-Bose-Shrikhande spaces Desarguesian affine spaces Hering plane of order 27 Nearfield plane of order 9 Two Hering designs with v = 3 6 and k = 3 2 Lüneburg planes

7 One-dimensional affine case Description G acts on a vector space GF(p) d, and Z d p G AΓL(1, p d ) AGL(d, p) Group theory is no longer relevant; field theory comes into play. W. M. Kantor 1993 Is there a hope for a complete classification of these designs? I suspect not. My evidence is the number of examples described below, together with their somewhat wild appearance.

8 Known examples Some translation affine planes. Generalised Netto systems. Inflations. Some constructions by Kantor. Munemasa s examples.

9 Theorem (Bamberg & Pauley 2006) If P is an irreducible polynomial over GF(q 2 ) of degree d such that x, y GF(q 2 ), x d P(x q 1 ) y d P(y q 1 ) GF(q) x y GF(q), (*) then there arises a flag-transitive linear space with a one-dim. affine aut. group, and q 2d points and line-size q 2. Projective permutation polynomials (Pauley-nomials) The map x x d P(x q 1 ) is a permutation of the projective line GF(q 2 )/GF(q).

10 Desarguesian affine spaces b GF(q 2m ) with b q+1 1, P is the minimal polynomial of b over GF(q 2 ), Desarguesian affine space b GF(q 2 ). Kantor s type IV example ζ is a generator of GF(q 2 ), d is an odd divisor of q 1, Then P(x) := x d ζ is irreducible and satisfies (*).

11 Proof. Not difficult to prove that P(x) is irreducible. Now for condition (*): x d P(x q 1 ) y d P(y q 1 ) GF(q) x y GF(q) x d ( x (q 1)d ζ ) y ( d y (q 1)d ζ ) = k GF(q) xqd ky qd = ζ(x d ky d ) (x d ky d ) q = ζ(x d ky d ) x d = ky d or ζ = (x d ky d ) q 1 (x/y) d GF(q) x/y GF(q)

12 Inflations P(x) P(x s ). Certain conditions on s etc. An infinite family Let p be an odd prime. Then P(x) := x p + x p x 1 is irreducible over GF(p) and satisfies the condition.

13 Proof P(x) := x p + x p x 1 Let z be a root of P. z 1 since P(1) = 1. z / GF(p) as otherwise P(z) = z 1 0. zp z 1 z 1 By induction so 2 = 0 zp = (2z 1)/z z pp z pi = = (i + 1)z i iz (i 1), (p + 1)z p pz (p 1) z z GF(p p ) and hence P is irreducible. Satisfies condition (*): an exercise.

14 Open problem Find more infinite families of Pauley-nomials: x d P(x q 1 ) y d P(y q 1 ) GF(q) x y GF(q)