ABSTRACT
To elaborate on the comments of Chapter 70 and to digress slightly, a nearfield plane is a translation plane of order k that admits a group of affine homologies of order k − 1. In this case, it follows directly that there are two affine homology groups of order k − 1. The basic question is how large must a homology group be before it can be concluded that one obtains, in fact, the full group? For example, suppose there is a homology group of order (k − 1)/2 or perhaps two such groups. Must then the plane be a nearfield plane? Actually, the answer is ‘no’; there are translation planes of order k that admit one or even two affine homology groups of order (k−1)/2. The complete answer when there are two groups is due to Hiramine and Johnson.
