The object of size 1

The object of size 1, stabilizer $G$ , one class of 1 subgroup · rigid

The one-point object: what the whole group fixes, the plane, the line, the curve, the graph and Kirmse’s lattice.

The object of size 1: the Fano plane, the projective line and the Coxeter graph themselves, each fixed by the whole group.

Its incarnations

The seam table names the object in each of the five theories that meet the group. Every entry there is built.

Fano plane: the plane
Projective line: the line
The group: subgroups 1 and $G$
Klein quartic: the curve
Graphs: the graph

The stabilizer is its own normalizer, so between any two incarnations there is exactly one seam, and the seams agree.

The fifteen objects

The object on the line of sizes. In ink, the objects it maps onto; in blue, those that map onto it. Dots count each object’s automorphisms, and dashed brackets join the pairs with one permutation character.

Example

Its incarnations are the figures fixed by the whole group: the Fano plane itself, the projective line, the subgroups 1 and $G$ , the Klein quartic, the Coxeter graph, the octonion algebra, and Kirmse’s lattice, the one lattice $\Z^8+\tfrac12C$ invariant under the collineations of the octonion table, which is not closed under multiplication.

Remark

It is rigid, and its row of the table of marks is 1 in every column.

Examplecomputed

It is also Thurston’s congruence link complement itself. Kirmse’s lattice, not closed under its own table, is closed under seven other multiplications on the same units: under the octonion product carried to a Fano plane $B$ , a lattice $L_A$ is closed exactly when $A$ and $B$ share three lines. So Kirmse’s lattice is an order after all, for seven multiplications, though not for its own.

Concepts: objectrigid objectGalois categorybridge
In the Esquisse: 5Courte marche à travers la théorie de Galois8La famille de Weyl9Immeubles et réseaux10La table en deux, en sept et à l’infini19Une formule du produit
The volume’s word: lift
In the volume: IThe Founding SentenceXVIIITwo Parents of the Sky