The object of size 42, class b

The object of size 42, stabilizer $V_4^b$ , one class of 7 subgroups · 6 automorphisms

Stabilizer $V_4^b$ and automorphism group $S_3$ : ordered pairs of points of the Fano plane, where natural seams match swaps with swaps and rotations with rotations.

The object of size 42 of class

b

as the ordered pairs of points of the Fano plane:

(2, 4)

on the line 246, and the rotation

(p,q)\mapsto(q,p+q)

that carries it to

(4, 6)

and

(6, 2)

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: ordered pairs of points
Projective line: pairs of disjoint pairs, cross-ratio $\{3,5\}$ , orbit of $\{\{0,1\},\{2,4\}\}$
The group: ordered pairs of commuting involutions generating a $V_4^b$
Klein quartic: ordered pairs of centres of two involutions generating a $V_4^b$
Graphs: Coxeter pairs at distance 4 with a common point

The seams between two incarnations form a torsor under $N_G(H)/H$ , a group of order 6, so there are 6 of them. The object the object of size 42, class a has the same permutation character, yet for one marking no seam joins the two.

Remark(marking)

Which class is which depends on the marking. The outer automorphism exchanges each class $a$ with its class $b$ , so the points and the lines of the Fano plane are assigned their classes only once the group of the plane is identified with the group of the projective line. The seam table and this journal’s labels use the identification of Seams, under which the points are class $a$ . The volume’s own chart, its table of observers and pairs in Chapter X, uses the other: there the stabilizer of the clock 1 fixes the bisection $\{0,1,2,4\}\mid\{3,5,6,\infty\}$ , which lies in class $b$ . Read in that chart, the names exchange: the clocks are class $b$ and the lines class $a$ , and so on for $A_4$ and $V_4$ .

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

$N_G(V_4^b)=S_4^b$ , so the automorphism group is $S_3$ . An ordered pair of points $(p,q)$ of the Fano plane determines the line $L=pq$ , and its stabilizer is the group of elations with axis $L$ , a member of $V_4^b$ .

Proposition(Swaps and rotations)

On ordered pairs $(p,q)$ of points, on ordered pairs $(a,b)$ of commuting involutions generating a group of class $V_4^b$ , and on ordered pairs of centres of such involutions, the swaps and the rotations

(p,q)\mapsto(q,p+q),\qquad(a,b)\mapsto(b,ab),\qquad(c_a,c_b)\mapsto(c_b,c_{ab})

generate the full automorphism group; a swap has quotient class $D_8$ and a rotation $A_4^b$ . The elation seam, sending $(p,q)$ to the elations with axis $pq$ and centres $p$ and $q$ , and the centre seam, sending $(c_a,c_b)$ to $(a,b)$ , carry swaps to swaps and rotations to rotations. The third vertex $c_{ab}$ of the rotation is the third vertex of the self-polar triangle.

Proof

The elations with a common axis $L$ form a group of class $V_4^b$ , with one nontrivial element for each centre on $L$ , and the product of the elations with centres $p$ and $q$ is the one with centre $p+q$ . The rest was checked by machine.

Example

Ordered pairs of points of the Fano plane; pairs of disjoint pairs of $\Proj^1(\F_7)$ with cross-ratio $\{3,5\}$ , in the orbit of $\{\{0,1\},\{2,4\}\}$ ; ordered pairs of commuting involutions generating a $V_4^b$ , and of their centres; and the pairs of Coxeter vertices at distance 4 whose antiflags have a common point.

Remark

It is the Gassmann partner of the object of class $a$ , and every bridge between them, for one marking, is refuted. A natural seam system on it with non-abelian monodromy is not known.

Example

In Thurston’s congruence link complement it is the tetrahedra of class $b$ with a pair of opposite edges.

Concepts: coherenceseam monodromyquotient classrefutedabsenceforced gapGalois gapimprint
In the Esquisse: 3La table des sutures du groupe d’ordre 1684La monodromie des sutures10La table en deux, en sept et à l’infini16Une loi de réciprocité17L’écart de Galois
The volume’s word: antipodeletter