Translation planes admitting a pair of index three homology groups (Q1610988)

The author studies translation planes \(A\) of finite order \(q\) that admit two distinct homology groups of order \((q-1)/3.\) This follows the lines of [\textit{Y. Hiramine} and \textit{N. L. Johnson}, Geom. Dedicata 43, No.~1, 17-33 (1992; Zbl 0764.51009)]. It is first shown that \(A\) admits two homology groups of order \((q-1)/3\) in the translation complement such that the center of every one group is on the axis of the other. The main theorem states that \(A\) is either a generalized André plane, or possibly \(q=2^6\) or \(q=p^2\) where \(p\) is from a ten element list of primes. It is left open whether the listed \(q\)'s really constitute exceptions, but the irregular nearfields with solvable multiplicative group show that exceptions do exist. In section 4 an interesting theorem on nonsolvable homology groups is presented.