A translation plane of order 49 with non-solvable collineation group
The author constructs a translation plane of order 49 such that the translation complement S has a normal subgroup F of order 24 and S mod F is isomorphic to the symmetric group \(S_ 5\). There is a normal subgroup of index 2 which induces the same orbit decomposition on \(1_{\infty}\) as does SL(2,5) on the Desarguesian affine plane of order 49, but this plane is not Desarguesian. The plane is a rank 3 plane with two orbits on \(1_{\infty}\) (under S) of respective lengths 30 and 20. The construction goes back to an idea of Bruen involving what he calls chains of circles. (The line at infinity of a Desarguesian affine plane of order \(q^ 2\) becomes a Miquelian inversive plane in which the ''circles'' are the derivation sets. The circles in Bruen's chains are not disjoint.) The plane of this paper appears not to belong to any class of known translation planes; in any case, no known translation plane has been given explicitly with a proof that the translation complement has the properties of this particular one. We have here an interesting exceptional case.
scientific article; zbMATH DE number 3879615
scientific article