Oval configurations of involutions in symmetric groups (Q1377788)

An oval configuration of involutions in the symmetric group \(S_n\), where \(n\) is even, is a set of fixed-point-free involutions of \(S_n\) with the property that, given any two disjoint transpositions \((a,b)\) and \((c,d)\) there is precisely one involution in the set which involves both \((a,b)\) and \((c,d)\). Let \({\mathcal O}=\{1,\dots,n\}\) be an oval of a projective plane of even order. If \(P\) is a point not on the oval, then all lines through \(P\) meeting the oval determine a fixed-point-free involution of \(S_n\). The set of involutions determined by all points of the plane not on the oval forms an oval configuration. All known examples of oval configurations, except for one (discovered by Mathon in \(S_{10}\)), arise via this construction from ovals. Theorem. There is no oval configuration of involutions in \(S_{14}\) invariant under conjugation by a Frobenius subgroup of order 39.