La table des sutures du groupe d’ordre 168

Les quinze classesThe fifteen classes

$G$ has exactly 179 subgroups, in fifteen conjugacy classes. With the number of conjugates and the quotient $N_G(H)/H$ they are: the trivial group (1; $G$), $C_2$ (21; $C_2\times C_2$), $C_3$ (28; $C_2$), $C_4$ (21; $C_2$), two classes of Klein four-groups $V_4^a$ and $V_4^b$ (7 each; $S_3$), $S_3$ (28; 1), $C_7$ (8; $C_3$), $D_8$ (21; 1), two classes $A_4^a$ and $A_4^b$ (7 each; $C_2$), the Frobenius group $7{:}3$ (8; 1), two classes $S_4^a$ and $S_4^b$ (7 each; 1), and $G$. So the fifteen objects have sizes 168, 84, 56, 42, 42, 42, 28, 24, 21, 14, 14, 8, 7, 7 and 1.

The labels $a$ and $b$ are fixed by the marking: $S_4^a$ is the class of the stabilizers of the points of the Fano plane and $S_4^b$ that of its lines, and $V_4^a$, $A_4^a$ (respectively $V_4^b$, $A_4^b$) are the normal Klein four-group and the alternating group of a member of $S_4^a$ (respectively $S_4^b$). In $\GL(3,2)$ a member of $V_4^a$ is the group of elations with a common centre and a member of $V_4^b$ the group of elations with a common axis; a member of $S_4^a$ contains its normal $V_4^a$ and three members of $V_4^b$, and dually.

The outer automorphism group has order 2, induced by conjugation by any element of $\PGL(2,7)$ outside $G$, such as $z\mapsto3z$. It fixes nine classes and exchanges the three pairs $(V_4^a,V_4^b)$, $(A_4^a,A_4^b)$ and $(S_4^a,S_4^b)$, so which member of a pair carries the label $a$ is a matter of marking: changing $\mu_A$ by an outer automorphism exchanges the labels.

Objets rigides et non rigidesRigid and non-rigid objects

By the rigidity criterion an object is rigid exactly when its stabilizer is self-normalizing. Six of the fifteen are: those with stabilizers $S_3$, $D_8$, $7{:}3$, $S_4^a$, $S_4^b$ and $G$, of sizes 28, 21, 8, 7, 7 and 1. For each of them all seams between incarnations are unique and consistent: once the markings are fixed, the antiflags, the flags, the points of the projective line, the points and the lines of the Fano plane, and the plane itself are matched across theories in exactly one way.

The other nine, of sizes 168, 84, 56, 42, 42, 42, 24, 14 and 14, have automorphism groups $G$, $C_2\times C_2$, $C_2$, $C_2$, $S_3$, $S_3$, $C_3$, $C_2$ and $C_2$. Between two incarnations of one of them there is more than one seam, and the seams that theories supply naturally need not agree.

Les marquesMarks

For subgroups $H$ and $K$, the mark of $K$ on the object $G/H$ is the number $m(H,K)=|(G/H)^K|$ of cosets that $K$ fixes; for a finite $G$-set $X$, its mark at $K$ is $|X^K|$. The term is Burnside’s. A mark depends only on the classes of $H$ and $K$, and marks add over disjoint unions.

The coset $gH$ is fixed by $K$ exactly when $K\le gHg^{-1}$, so a mark counts the conjugates of $H$ that contain $K$, each $|N_G(H):H|$ times. Ordered by size, the table is lower triangular with the automorphism counts on the diagonal, and so invertible: the marks of a finite $G$-set give its numbers of orbits of each type, and with them the set. For the group of order 168 the row of $G/S_4^a$ reads 7,3,1,1,1,3,1,0,1,1,0,0,1,0,0: the seven points of the Fano plane, of which an involution fixes three, an element of order 3 one, and an element of order 7 none.

The permutation character of $G/H$ is its row read at the cyclic subgroups, the columns 1, $C_2$, $C_3$, $C_4$ and $C_7$. Burnside’s marks, read at every subgroup, are a finer invariant, and they determine a finite $G$-set up to isomorphism.

Les paires de GassmannThe Gassmann pairs

Read along its cyclic columns, the table gives the permutation characters, and three pairs of objects share theirs: $(G/V_4^a,G/V_4^b)$, $(G/A_4^a,G/A_4^b)$ and $(G/S_4^a,G/S_4^b)$. The reason is conceptual. The outer automorphism exchanges the two members of each pair and fixes every conjugacy class of elements except $7A$ and $7B$; $V_4$, $A_4$ and $S_4$ contain no element of order 7, so the two members meet every class of $G$ in equally many elements, which is the condition for equal permutation characters. Since the outer automorphism sends $7A$ to $7B=7A^{-1}$, this is an instance of the Galois gap of Chapter 17: counting cannot hear a twist that acts on the classes as Galois acts.

The column $V_4^a$ separates them: within each pair its marks differ, 6 and 0 on the objects of size 42, 2 and 0 on those of size 14, 1 and 3 on the sevens. By Burnside’s theorem the two members of a pair are not isomorphic, no seam joins them, and every bridge between them, for one marking, is refuted. In the language of Chapter 2 each pair is a separating absence whose imprint is the column $V_4^a$, which separates all three at once. The pair of $S_4$‘s is the classical one, the points and the lines of the Fano plane.

Le premier blocThe first block

In each of five theories take natural sets of figures, let the theory’s group act through its marking, split the sets into orbits, and compute the stabilizer class of each orbit. An orbit is then an incarnation of the object of that class, and two orbits with the same class are joined by the explicit seam $gx\mapsto gy$ for points with equal stabilizers, so every entry of the table has status built. The markings are $\mu_A$ for the Fano plane and the graphs, the identity for the projective line and the group, and Klein’s $\rho$ for the quartic.

On the projective line the figures are subsets, pairs of pairs and vectors. The 8 points are one orbit, of class $7{:}3$; the 28 pairs one orbit, $S_3$; the 56 three-subsets one orbit, $C_3$. The 70 four-subsets split into three: 42 in the orbit of $\{0,1,2,3\}$, class $C_4$, and 14 each in the orbits of $\{0,1,2,5\}$ and $\{0,1,2,4\}$, classes $A_4^a$ and $A_4^b$. The 35 bisections, partitions into two sets of four, split as $21+7+7$, classes $D_8$, $S_4^a$ and $S_4^b$. The 24 nonzero vectors of $\F_7^2$ up to sign, on which $G$ acts through $\SL(2,7)$, are the object of size 24, and the ordered triples form two regular orbits, of $(0,1,2)$ and of $(0,1,3)$.

The other columns are as concrete. In the Fano plane: frames, flags and antiflags, ordered pairs of points and of lines, oriented quadrangles and quadrilaterals, and cyclic labellings, bijections to $\Z/7$ carrying the lines onto the translates of $\{0,1,3\}$ or of $\{0,4,6\}$. In the group: elements and subgroups under conjugation, and pairs of commuting involutions. On the Klein quartic: its flexes and their tangents, the bitangents with their poles and points of contact, the centres and axes of the involutions, self-polar triangles, the eight flex triangles, and two families of invariant conics. In the graphs: Coxeter vertices, arcs, edges and heptagons, and Heawood edges, hexagons and perfect matchings.

Sutures internes à une théorieSeams inside one theory

Some relations between entries are seams that hold for every marking, because they are made from the figures alone. 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 off the line, which is a point of the line: undirected 4-cycles are flags. Directing the cycle halves the stabilizer, $D_8$ becoming $C_4$.

Each $S_3$-orbit of perfect matchings of $\Proj^1(\F_7)$ contains a distinguished pair, the one its stabilizer fixes, and that is its seam to the pairs; the two 7-orbits of matchings are the two families of $K_4$‘s inside the twenty-eight. On the quartic, the stabilizer of a bitangent is an $S_3$ whose subgroup of order 3 fixes both points of contact and whose involutions exchange them, the stabilizer of a point of contact being only $C_3$. So a point of contact is a bitangent together with one of its two points of contact, and sending it to its bitangent is the natural $G$-map $G/C_3\to G/S_3$, with fibres of size two. Each centre lies on exactly four bitangents, which the two elements of order 4 fixing it pair into the two Coxeter edges there.

Cases vides par nécessitéCells that are empty by necessity

Every row of the Fano, projective-line and Klein columns is filled, but inside a column particular kinds of figure can realize only some classes, and these restrictions are theorems. A conjugacy class of elements is an incarnation of $G/C_G(x)$, and the centralizers are $G$, $D_8$, $C_3$, $C_4$, $C_7$, $C_7$; a class of subgroups is an incarnation of $G/N_G(H)$, and the normalizers are $G$, $D_8$, $S_3$, $7{:}3$, $S_4^a$, $S_4^b$. So the objects with stabilizers 1, $C_2$, $V_4^a$, $V_4^b$, $A_4^a$, $A_4^b$ are incarnated by no conjugacy class at all.

Under Klein’s $\rho$ no point of $\Proj^2(\C)$ has its stabilizer in the classes $V_4^a$, $V_4^b$, $A_4^a$, $A_4^b$, $7{:}3$, $S_4^a$, $S_4^b$ or $G$. This absence has an imprint, the self-polar triangle: the object $G/V_4$, which no point can carry, is carried by ordered pairs of vertices of a self-polar triangle. A point of the Klein quartic itself has stabilizer 1, $C_2$, $C_3$ or $C_7$, with orbits of 84, 56 and 24 points beside the regular ones.

Two further families come from the literature. Klein’s map of type $\{7,3\}$ on the quartic has 24 heptagonal faces, 56 vertices and 84 edges, and $G$ acts by rotations about their centres: the face centres are the flexes, the vertices are the points of contact of the bitangents, and the midpoints of the edges are the points with stabilizer $C_2$. The 24 cusps of the modular curve $X(7)$, the preimage of the cusp of $X(1)$, are the object of size 24 again, and return in Chapter 4.