Semifield planes with a transitive autotopism group (original) (raw)

A characterization of the generalized twisted field planes of characteristic ?5

Geometriae Dedicata, 1994

Let 7r be a non-Desarguesian Semifield plane of order p~, p a prime number > 5 and n _> 3, and let G denote the group induced by the autotopism group G of rr on the line at infinity. We prove that 7r is a generalized twisted field plane if, and only if, G has an element of order (pk _ 1)((pn _ 1)/(pro _ 1)), for some integers k and ra, where k [ m, m ] n, and ra < n.

Semifield planes of order p 4 and kernel GF(p 2)

Journal of Geometry, 2005

In this article we determine the number of non-isomorphic semifield planes of order p 4 and kernel GF(p 2) for p prime, 3 ≤ p ≤ 11. We show that for each of these values of p, the plane is either desarguesian, p-primitive, or a generalized twisted field plane. We also show that the class of p-primitive planes is the largest. We also discuss the autotopism group of the semifields under study.

Plane curves with a large linear automorphism group in characteristic ppp

2022

Let GGG be a subgroup of the three dimensional projective group mathrmPGL(3,q)\mathrm{PGL}(3,q)mathrmPGL(3,q) defined over a finite field mathbbFq\mathbb{F}_qmathbbFq of order qqq, viewed as a subgroup of mathrmPGL(3,K)\mathrm{PGL}(3,K)mathrmPGL(3,K) where KKK is an algebraic closure of mathbbFq\mathbb{F}_qmathbbFq. For the seven nonsporadic, maximal subgroups GGG of mathrmPGL(3,q)\mathrm{PGL}(3,q)mathrmPGL(3,q), we investigate the (projective, irreducible) plane curves defined over KKK that are left invariant by GGG. For each, we compute the minimum degree d(G)d(G)d(G) of GGG-invariant curves, provide a classification of all GGG-invariant curves of degree d(G)d(G)d(G), and determine the first gap varepsilon(G)\varepsilon(G)varepsilon(G) in the spectrum of the degrees of all GGG-invariant curves. We show that the curves of degree d(G)d(G)d(G) belong to a pencil depending on GGG, unless they are uniquely determined by GGG. We also point out that GGG-invariant curves of degree d(G)d(G)d(G) have particular geometric features such as Frobenius nonclassicality and an unusual variation of the number of mathbbFqi\mathbb{F}_{q^i}mathbbFqi-rational points. F...

Projective planes with a transitive automorphism group

In this note we prove two theorems which contribute towards the clas-sification of line-transitive designs. A special class of such designs are the projective planes and it is this problem which the paper addresses. There two main results:-Theorem A: Let G act line-transitively on a projective plane P and let M be a minimal normal subgroup of G. Then M is either abelian or simple or the order of the plane is 3, 9, 16 or 25. Theorem B: Let G be a classical simple group which acts line-transitively on a projective plane. Then the rank of G is bounded.

On automorphisms of semifields and semifield planes

2018

Изучается взаимосвязь полуполевой проективной плоскости и ее координатизирующего полуполя с использованием линейного пространства и регулярного множества. Установлен геометрический смысл инволюторного автоморфизма конечного полуполя, доказаны некоторые его свойства

On the semifield planes of order and dimension 2 over the kernel

Note di Matematica, 2003

In this article we consider the problem of determining all non-Desarguesian semifield planes of order 5 4 and kernel GF(5 2). We show that the class of p-primitive planes is the largest class and besides those the only other semifield planes in the class under study are the generalized twisted field planes. We conjecture that in general these two classes include all the non-Desarguesian semifield planes of order p 4 and kernel GF(p 2).

On alternating subgroup A_5A_5A_5 in autotopism group of finite semifield plane

2020

We discuss well-known hypothesis that the full collineation group of any finite non-Desarguesian semifield plane is solvable. We continue to investigate the semifield planes of odd order which admit an autotopism subgroup isomorphic to alternating group A5. It is proved that a semifield plane of any odd order does not admit A5 in autotopism group.

Semifield planes of order with kernel and center

European Journal of Combinatorics, 2006

A classification of semifield planes of order q 4 with kernel F q 2 and center F q is given. For q an odd prime, this proves the conjecture stated in [M. Cordero, R. Figueroa, On the semifield planes of order 54 and dimension 2 over the kernel, Note Mat. (in press)]. Also, we extend the classification of semifield planes lifted from Desarguesian planes of order q 2 , q odd, obtained in [M. Cordero, R. Figueroa, On some new classes of semifield planes, Osaka J. Math. 30 (1993) 171-178], to the even characteristic case.