Translation planes of order \(q^ 3\) which admit SL(2,q) (Q1074859)

The authors study the translation planes of order \(q^ 3\) with kernel \(K\supseteq GF(q)\), admitting a collineation group \(G\simeq SL(2,q)\). They remark that G must be completely reducible and prove the following: If the underlying vector space is written in the form \(V_ 2\oplus V_ 2^{\lambda}\oplus V_ 2^{\mu}\), where \(\lambda\), \(\mu\) are automorphisms of GF(q), then the equations \(xx^{\lambda}=1\), \(xx^{\mu}=1\) and \(x^{\lambda}x^{\mu}=1\) have no solutions in GF(q) different from \(\pm 1\). G has on the line at infinity two orbits if q is even and three orbits if q is odd. In case \(\lambda =1=\mu\), the plane is desarguesian. For \(\lambda\) \(\neq 1\neq \mu \neq \lambda\) an example is given, which constitutes a new class of nondesarguesian translation planes for \(q=p^{3\omega}\), where p is an arbitrary prime. The even order planes of this class are not isomorphic to any Kantor plane [\textit{W. M. Kantor}, SIAM J. Algebraic Discrete Methods 3, 151-165 (1982; Zbl 0493.51008)].