Michael J. Kallaher

Collineation groups irreducible on the components of a translation plane

AbstractWe investigate finite translation planes of odd dimension over their kernels in which the translation complement induces on each component l a permutation group whose order is divisible by a p-primitive divisor. Using results of this investigation, we show that rank 3 affine planes of odd dimension over their kernels are either generalized André planes or semi-field planes. A similar result is given for translation planes having a collineation group which is doubly transitive on each affine line; besides the above two possibilities, there is a third possibility; the plane has order 27, the translation complement is doubly transitive on

SIMPLE GROUPS ACTING ON TRANSLATION PLANES

\mathfrak{U}

A conjecture on semi-field planes. II

, and SL(2, 13) is contained in the translation complement.We also consider translation planes of odd dimension over their kernels which have a collineation group isomorphic to SL(2, w) with w prime to 5 and the characteristic, and having no affine perspectivity. We show that such planes have order 27, the prime power w=13, and the given group together with the translations forms a doubly transitive collineation group on {ie153-1}. This indicates quite strongly that the Hering translation plane of order 27 is unique with respect to the above properties.

Collineation groups whose order is divisible by a p-primitive divisor

Publisher Summary This chapter focuses on translation planes with emphasis on the classes of translation planes, dimensions and collineations, and collineation groups. A translation plane is an affine plane II whose translation group T (II) is (sharply) transitive on the affine points. (By a translation is meant a (P, l∞)-elation, where l∞ is the line at infinity, or improper line, of II and P ϵ l∞.) Every coordinatizing ternary ring of a translation plane (with respect to the line l∞ ) is quasified. A generalized Hall system is a quasifield derived from a semifield, and a generalized Hall plane is a translation plane coordinatized by a generalized Hall system. It the semifield is itself a field, that is the plane is Desarguesian, then the derived quasifield is called a Hall system, and the derived plane a Hall plane. A planar collineation of a translation plane II is a collineation which fixes a subplane pointwise. A planar collineation group of II is a collineation group which fixes a sub- plane pointwise.

Half-Bol quasi-fields

We discuss the possibility of finite simple groups acting as collineation groups on finite translation planes of odd order with special attention paid to the sporadic simple groups. We assume such a group acts irreducibly (in the vector space sense) on the plane. It is shown that if the characteristic of the plane does not divide the order of the group, then the group cannot be one of eleven sporadic simple groups. Also, if one of the Mathieu groups acts irreducibly on a finite translation plane then it is either M11 or M23.

A note on planes of characteristic three

Many special cases of (*) have been settled. For instance, Cohen et al [1] have shown that if G fixes two non-zero points of V, that do not both lie in the same component of T, then T is the spread associated with either a Hall plane or the Lorimer-Rahilly plane of order 16 (LR-16) [14], [18]. Another such result is given in [8] ; there it is shown that if g is a prime number and G is a one-dimensional projective unimodular group then r is the spread associated with one of the following translation planes: (1) the Desarguesian planes of order 4, 8, or 9; (2) the nearfield plane of order 9; (3) LR-16; (4) the translation plane JW-16, obtained by transposing the slope maps of LR-16 [19]. The purpose of this article is to study (*) when G is a group of type PSL(n, w) when n ^ 3. Our object will be to show that under these conditions T is the spread associated with LR-16 or JW-16 and G is PSL(3, 2). Thus, in terms of translation planes, our result may be stated as follows.

Solvable, flag-transitive, rank 3 collineation groups

ConclusionThe results in the previous sections lend strong support to the conjecture made in the Introduction. Furthermore, if the long-standing conjecture concerning the solvability of autotopism groups for semi-field planes is correct then the probability of our conjecture being true is greatly increased. In any case the existence of a semi-field plane π for which u(π) = 2, 3, or 4 would provide a counterexample to the earlier conjecture.There are examples of semi-field planes with u(π) = %. As mentioned in Example 2 of Section 3, one of the semi-field planes of order 16 has u(π)-5. For that plane, the five orbits of the autotopism group G in ϕ(G) have lengths 27, 36, 54, 54, 54. The union of the orbit having length 36 and one of those having 54 is the union of the points in ϕ(G) on 6 lines through a vertex U and the union of the remaining three orbits consists of the 135 points on the remaining 9 lines through U. There are also non-Desarguesian A-planes in which u(π) = 5; the semi-field plane of order 34 coordinatized by the twisted field of Albert has u(π) = 5.

Fixed point free linear groups, rank three planes, and Bol quasifields

SummaryLet π be a translation plane of order q3 with kernel GF(q). Our main result is that the translation complement of π cannot contain a group G such that G/Z(G)=A7. This removes a possible exception to the results in our paper “Collineation groups irreducible on the components of a translation plane.”We also show that the assumptions of the above paper can be relaxed slightly.