Translation planes of odd order and odd dimension

T G Ostrom

doi:10.1155/s0161171279000181

Abstract

The author considers one of the main problems in finite translation planes to be the identification of the abstract groups which can act as collineation groups and how those groups can act.The paper is concerned with the case where the plane is defined on a vector space of dimension 2d over GF(q), where q and d are odd. If the stabilizer of the zero vector is non‐solvable, let G0 be a minimal normal non‐solvable subgroup. We suspect that G0 must be isomorphic to some SL(2, u) or homomorphic to A6 or A7. Our main result is that this is the case when d is the product of distinct primes.The results depend heavily on the Gorenstein‐Walter determination of finite groups having dihedral Sylow 2‐groups when d and q are both odd. The methods and results overlap those in a joint paper by Kallaher and the author which is to appear in Geometriae Dedicata. The only known example (besides Desarguesian planes) is Hering′s plane of order 27 (i.e., d and q are both equal to 3) which admits SL(2, 13).

Highlights

The author considers one of the main problems in finite translation planes to be the identification of the abstract groups which can act as col lineation groups and how those groups can act
The methods and results overlap those in a joint paper by Kallaher and the author which is to appear in Geometriae Dedicata
Qd A translation plane of order with kernel F GF(q) may be represented by a vector space of dimension 2d over F. (The plane is usually said to have dimension d over F.) Here the points are the elements of the vector space and the lines are the translates of the components of a spread

Summary

INTRODUCTION

Qd A translation plane of order with kernel F GF(q) may be represented by a vector space of dimension 2d over F. (The plane is usually said to have dimension d over F.) Here the points are the elements of the vector space and the lines are the translates of the components of a spread. The Sylow 2-groups of the induced permutation group on are cyclic or dihedral (when dimension and order are odd); it is possible that the key non-solvable group is always SL(2,u) for some u or, perhaps, is a pre-image of A6 or A7. This is suggested by the Gorenstein-Walter Theorem [5]. Our most important result is Theorem (3.5) which states that, if d is the product of distinct primes, a minimal non-solvable normal subgroup of the linear translation complement either has the form SL(2,u) or is a pre-image of A6 or A7. A normal subgroup of a linear group G is a minimal non-f.p.f, group with respect to G if it is not fixed point free but every normal subgroup of G properly contained in it is f.p.f

An irreducible group of linear transformations acting on a vector space

Then T

REMARK Except for the consideration of the possibilities that the

IGoI If p divides