Six-point matching geometry

The complete graph on six points has fifteen edges and fifteen perfect matchings. Its six one-factorizations organize a dual incidence geometry exchanging points with factorizations and edges with matchings. The classical names are duads, synthemes and totals of synthemes. This geometry supports duad-syntheme duality on six points and a construction of the small Witt design.

The six pentads are the six one-factorizations of the complete graph on six points. Permuting the ground-set points permutes these factorizations. The action sends a transposition to three disjoint transpositions and a $3$-cycle to two disjoint $3$-cycles. Its restriction to $A_6$ is faithful by simplicity, and the full action of $S_6$ is faithful because an order-two kernel would be central. It is consequently an automorphism of $S_6$. Its restriction preserves $A_6$ and changes $3$-cycle type, so it cannot be induced on $A_6$ by any conjugation from $S_6$.

A bijection from the points of one six-set to the totals of another extends by incidence to bijections from duads to synthemes, synthemes to duads, and totals to points. A duad maps to the intersection of its two point-totals. The same duality exchanges internal duads of a three-versus-three partition with cross synthemes of its corresponding partition. The latter rule follows by splitting the six cross perfect matchings into parity classes of three, yielding two triangles on the six totals.

A duad is an unordered pair of distinct points. On a six-point set there are $\left(\genfrac{}{}{0ex}{}{6}{2}\right)=15$ duads. They are the edges of the complete graph and are grouped into synthemes.

A syntheme partitions six points into three duads, equivalently a perfect matching of the complete graph. There are $6!/(2^{3}3!)=15$ synthemes. Every duad belongs to three. Two edge-disjoint synthemes extend to a unique total of synthemes: their union is a six-cycle, whose complement has a unique factorization into three matchings.

A total is a set of five edge-disjoint synthemes covering all fifteen duads of a six-point set. There are six totals; each syntheme belongs to two and every two totals share exactly one syntheme. Each total contains exactly one syntheme containing any prescribed duad. Totals are one-factorizations of the complete graph.

Two disjoint synthemes form a six-cycle, whose complement is a triangular prism. The prism has a unique partition into three perfect matchings, so the pair extends uniquely to a total of synthemes. Eight synthemes are disjoint from a fixed syntheme; grouping them four per total yields two totals through it, and double counting gives six totals in all.

// Target: combinatorics.bigb