The object of size 42, class a

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

Stabilizer $V_4^a$ and automorphism group $S_3$ : ordered pairs of lines of the Fano plane; it shares its permutation character with the class-b object and is not that object.

The object of size 42 of class

a

as the ordered pairs of lines of the Fano plane: the line 246, then the line 123, meeting at the point 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 lines
Projective line: pairs of disjoint pairs, cross-ratio $\{3,5\}$ , orbit of $\{\{0,1\},\{2,5\}\}$
The group: ordered pairs of commuting involutions generating a $V_4^a$
Klein quartic: ordered pairs of centres of two involutions generating a $V_4^a$
Graphs: Coxeter pairs at distance 4 with a common line

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 b 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

$N_G(V_4^a)=S_4^a$ , so the automorphism group is $S_3$ , not abelian, and monodromy is defined only up to conjugation. In $\GL(3,2)$ a member of $V_4^a$ is the group of elations with a common centre.

Example

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

Proposition(The Gassmann pairs of PSL(2,7))

It has the same permutation character as the object of class $b$ , and the two are not isomorphic, since their marks at $V_4^a$ are 6 and 0. Every bridge between them, for one marking, is refuted; the outer automorphism exchanges them.

Remark

No conjugacy class of elements or of subgroups is an incarnation of it, and no point or line of the complex projective plane has stabilizer $V_4$ under Klein’s representation: the fixed points of a Klein four-group are the centres of its involutions, with stabilizer $D_8$ . The object is carried instead by ordered pairs of vertices of a self-polar triangle.

Example

In Thurston’s congruence link complement it is the tetrahedra of class $a$ with a pair of opposite edges: the stabilizer of such a pair is the normal Klein four-group of the stabilizer $A_4$ of the tetrahedron.

Concepts: seam monodromyrefutedabsenceforced gapGalois gapimprint
In the Esquisse: 3La table des sutures du groupe d’ordre 16810La table en deux, en sept et à l’infini16Une loi de réciprocité17L’écart de Galois
In the volume: XIRulial Relativity