Elementary abelian 2-groups on the line at infinity of translation planes

Abstract

Let G be a subgroup of the linear translation complement of a translation plane of order qd with kernel GF(q) and let ¯G be the factor group modulo the scalars. We show that if ¯G is elementary abelian of order 2a, and if each involution in ¯G has a conjugate class of length greater than a+1 then 2e divides d, where e=[1/2(a+1)]−1. We show that one of Walker's planes is a counterexample if we drop the condition on lengths of conjugate classes. The Walker plane in question turns out to be of rank 3. This is one of Walker's planes of order 25 and was not previously known to have rank 3.