On Translation Planes which Admit Solvable Autotopism Groups Having a Large Slope Orbit

Our main object is to prove the following result.THEOREM C. Let A be an affine translation plane of order qr ≧ q2 suchthatl∞, the line at infinity, coincides with the translation axis of A. Suppose G is a solvable autotopism group of A that leaves invariant a set Δ of q + 1 slopes and acts transitively on l∞ \ Δ.Then the order of A is q2.An autotopism group of any affine plane A is a collineation group G that fixes at least two of the affine lines of A; if in fact the fixed elements of G form a subplane of A we call G a planar group. When A in the theorem is a Hall plane [4, p. 187], or a generalized Hall plane ([13]), G can be chosen to be a planar group.

On elations in semi-transitive planes

Letπbe a semi-transitive translation plane of even order with reference to the subplaneπ0. Ifπadmits an affine elation fixingπ0for each axis inπ0and the order ofπ0is not2or8, thenπis a Hall plane.

The existence of Fano subplanes in generalized Hall planes

One of the best known classes of non-Desarguesian planes is the calss of Hall planes (see Hall [2]). In [6] Hanna Neumann showed that the finite Hall planes of old order possess subplanes of order two (i.e., Fano subplanes)1. Kirkpatrick [5] has considered a type of plane which is generalization of the Hall planes and which he calls generalized Hall planes. In this paper we will give a sufficient condition that a finite generalized Hall plane possesses Fano subplanes. Some examples of odd order planes to which the condition applies shall be exhibited.

A note on generalized Hall planes

We prove that if π is a generalized Hall plane of odd order with associated Baer subplane π0 then π is a Hall plane if and only if there is a collineation σ of π such that π0σ ∩ π0 is an affine point.

A characterization of generalized Hall planes

We prove that a translation plane π of odd order is a generalized Hall plane if and only if π is derived from a translation plane of semi-translation class 1–3a. Also, a derivable translation plane of even order and class 1–3a derives a generalized Hall plane. We also show that the generalized Hall planes of Kirkpatrick form a subclass of the class of planes derived from the Dickson semifield planes.