ABSTRACT
In 1954, Hanna Neumann [1027] showed, using coordinates, that in any ‘projective’ Hall plane of odd order, there is always a projective subplane of order 2-a Fano plane. We recall that, of course, a Hall plane of order q2 may be characterized as a translation plane obtained from a Desarguesian affine plane of order q2 by the replacement of a regulus net and so is subregular. Here now the question is which subset of subregular planes of odd order must admit Fano configurations. If one considers the maximal possible cardinality set of q− 1 disjoint regulus nets, then a multiple derivation of all of these nets leads to another Desarguesian plane. Hence, it is not true that any subregular plane admits Fano configurations. However, we show that if t is less than or equal to roughly a quarter of the possible (q− 1), then such planes always admit Fano configurations.
