Spreads in admitting several homology groups of order

Abstract

The set of translation planes with spreads in admitting at least three homology groups with distinct axes of order is completely determined. Apart from the Desarguesian and Hall planes of order , the only possible plane is the Heimbeck plane of order admitting several quaternion homology groups of order 8. A classification is also given of all translation planes with spreads in that admit at least three distinct homology groups of order . Recent results conneting translation planes with spreads in admitting cyclic affine homology groups of order with conical flocks spreads provide the background for applications showing how the associated collineation groups are interrelated.