Some translation planes of order \(7^ 2\) which admit \(SL_ 2(9)\) (Q1059851)

Consider the hypothesis: \(\pi\) is a translation plane of dimension 2 over its kernel; further the translation complement has a (normal) subgroup K such that K/Z(K) is isomorphic to \(A_ 6.\) The author proves that there are exactly two isomorphism classes of planes \(\pi\) with kernel GF(7) which satisfy the hypothesis. If G is the translation complement of \(\pi\) and D the kernel of \(\pi\), then in one case G/D is isomorphic to \(A_ 6\) and in the second case, G/D is isomorphic to \(\Sigma_ 6.\) A comparison of this paper and that by the author and \textit{T. Ostrom} [ibid., 307-322 (1985; Zbl 0566.51020)] will be interesting.