stabilizer class

Floor 1, L’incarnation · introduced in Chapter 1, Un objet, plusieurs noms

What single datum decides which object a set is?

The conjugacy class formed by the stabilizers of an object’s points; it determines the object up to isomorphism.

Plate 1.2The stabilizer of the pair

\{0,\infty\}

: the maps

z\mapsto\lambda z

and

z\mapsto-\lambda/z

with

\lambda\in\{1,2,4\}

, a group

S_3

Definition(Stabilizer class)

The stabilizer class $\st(X)$ of an object $X$ is the conjugacy class formed by the stabilizers of its points.

It is a single class: $G_{gx}=gG_xg^{-1}$ , since $hgx=gx$ exactly when $g^{-1}hg\in G_x$ , and every conjugate occurs because $X=Gx$ .

Theorem(The stabilizer principle)

Two objects are isomorphic if and only if they have the same stabilizer class, and $X\mapsto\st(X)$ is a bijection from isomorphism classes of objects of $G$ onto conjugacy classes of subgroups of $G$ , with inverse $[H]\mapsto G/H$ .

Proof

An isomorphism $f$ has $G_{f(x)}=G_x$ , so the classes agree. Conversely, if they agree, there is $y\in Y$ with $G_y=G_x$ , and $gx\mapsto gy$ is a well-defined $G$ -isomorphism. The object $G/H$ has stabilizer class $[H]$ .

Example

Five sets of 28 elements on which the group of order 168 acts transitively all have stabilizers $S_3$ : the two-element subsets of $\Proj^1(\F_7)$ , where the stabilizer of $\{0,\infty\}$ is $\{z\mapsto\lambda z,\ z\mapsto-\lambda/z:\ \lambda\in\{1,2,4\}\}$ ; the Sylow 3-subgroups under conjugation, with stabilizer $N_G(P)$ ; the antiflags $(p,L)$ of the Fano plane, with stabilizer $\GL(p)\times\GL(L)\cong\GL(2,2)$ ; the bitangents of the Klein quartic, where the stabilizer of $x+y+z=0$ is $\langle h,s\rangle$ ; and the vertices of the Coxeter graph.

The 56 elements of order 3 form one conjugacy class, each with a centralizer of order 3, so $G$ has 28 Sylow 3-subgroups, and every subgroup of order 6 is the normalizer of its unique subgroup of order 3. So the subgroups of order 6 form a single class, the five sets have one stabilizer class, and they are one object.