The object of size 42, cyclic

The object of size 42, stabilizer $C_4$ , one class of 21 subgroups · 2 automorphisms

Stabilizer $C_4$ and automorphism group $C_2$ : the edges of the Coxeter graph, the directed 4-cycles on the quadrangles of the Fano plane, and the imaginary points of $\Proj^1(\F_{49})$ , whose automorphism is the Frobenius at 7.

The object of size 42 with stabilizer

C_4

as the 42 edges of the Coxeter graph; one edge, at

d_0

, in gold.

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: quadrangles with a directed 4-cycle
Projective line: harmonic pairs of disjoint pairs; 4-subsets in the orbit of $\{0,1,2,3\}$
The group: elements of order 4
Klein quartic: eigenvectors for $i$ of elements of order 4
Graphs: edges of the Coxeter graph

The seams between two incarnations form a torsor under $N_G(H)/H$ , a group of order 2, so there are 2 of them.

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.

Proposition

$N_G(C_4)=D_8$ , so the automorphism group is $C_2$ and the power is $\pm1$ . A directed 4-cycle on a quadrangle of the Fano plane goes to the element of order 4 advancing it one step, a seam onto the class $4A$ , and reversing the cycle corresponds to inversion.

Example

Quadrangles of the Fano plane with a directed 4-cycle; harmonic pairs of disjoint pairs of $\Proj^1(\F_7)$ , and the four-subsets in the orbit of $\{0,1,2,3\}$ ; the elements of order 4; the eigenvectors for $i$ of elements of order 4 in Klein’s representation; and the edges of the Coxeter graph.

Remark(Seams inside one theory)

A quadrangle is the complement of a line, and an undirected 4-cycle on it has its two diagonals in one parallel class of the affine plane, which is a point of the line: undirected 4-cycles are flags. Directing the cycle halves the stabilizer, and $D_8$ becomes $C_4$ . In the Coxeter graph, reversing an arc has quotient class $C_4$ , and the quotient is the set of edges.

Proposition(Imaginary points) computed

Let $\F_{49}=\F_7[\iota]$ with $\iota^2=-1$ . The 42 points of $\Proj^1(\F_{49})$ outside $\Proj^1(\F_7)$ , the imaginary points, form one orbit with stabilizer class $C_4$ ; each element of order 4 fixes exactly two of them, $z$ and $\bar z=z^7$ , acting there with multipliers $\iota$ and $-\iota$ . The Frobenius $z\mapsto z^7$ is the nontrivial automorphism of this incarnation. In the others it is inversion on $4A$ ; reversal on the directed 4-cycles; on a harmonic pair of pairs, the passage to the complementary four points, paired by the same involution; complement on the four-sets of points and of cusps; from the eigenvector for $i$ to the eigenvector for $-i$ of the same element; and on the Coxeter edges $\{(p,B),(q,B')\}\mapsto\{(p,B'),(q,B)\}$ .

The groups of order 8 of $\SL(2,7)$ generated by lifts of the elements of order 4 are its non-split tori, and for the non-split torus the object is the imaginary points, its automorphism the Frobenius at 7. In Thurston’s congruence link complement the object is the sets of four cusps that span no tetrahedron, each containing exactly one Coxeter edge.

Remark

The Coxeter graph, between the Fano plane and the projective line, compares their seams. On each edge the point rule and the line rule give mutually inverse elements of order 4, and the bracket rule, the square class of $[a,c][c,b][b,a]$ on the harmonic pair of pairs, agrees with the point rule exactly when the marking is in the class of $\mu_A$ . So the seam system of the vertex seam, the bracket rule and the point rule is coherent for that class and has the nontrivial automorphism as monodromy for the other.

Concepts: markingincarnationseam systemseam monodromypowerquotient classforced gap
In the Esquisse: 4La monodromie des sutures5Courte marche à travers la théorie de Galois10La table en deux, en sept et à l’infini11Le revêtement double et le miroir
The volume’s word: meeting
In the volume: XIIThe Coxeter GraphXXIIOne Speed