object

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

What is the one thing that several theories name?

A transitive set of a group; up to isomorphism, a conjugacy class of its subgroups.

Plate 1.1The fifteen objects of the group of order 168, one for each conjugacy class of subgroups, labelled by size and stabilizer; the object of size 28 is selected.

Definition(Object)

An object of a group $G$ is a transitive $G$ -set: a set $X$ with a left action of $G$ such that $X=Gx$ for some, equivalently every, $x\in X$ . For a subgroup $H\le G$ the cosets $G/H$ form an object, and the stabilizer of the coset $H$ is $H$ .

The word records the role these sets play: an object is what several theories name.

Theorem(The stabilizer principle)

Let $X$ and $Y$ be objects of $G$ , let $x\in X$ and $H=G_x$ .

(a) Evaluation at $x$ is a bijection from $\Iso_G(X,Y)$ onto $Y_H=\{y\in Y: G_y=H\}$ ; the isomorphism with value $y$ is $gx\mapsto gy$ .

(b) $X$ and $Y$ are isomorphic if and only if $\st(X)=\st(Y)$ . The assignment $X\mapsto\st(X)$ is a bijection from isomorphism classes of objects onto conjugacy classes of subgroups of $G$ , with inverse $[H]\mapsto G/H$ .

(d) If $X\cong Y$ , then $\Iso_G(X,Y)$ is a torsor for $\Aut_G(X)$ acting by precomposition, and for $\Aut_G(Y)$ acting by postcomposition. In particular it has $|N_G(H):H|$ elements.

Proof

(a) Since $X=Gx$ , a $G$ -map $f$ is determined by $f(x)$ , and an isomorphism has $G_{f(x)}=G_x$ . Conversely, if $G_y=H$ , then $f(gx)=gy$ is well defined, since $gx=g'x$ gives $g^{-1}g'\in H=G_y$ ; it is a $G$ -map, onto because $Y$ is transitive, and one-to-one because $gy=g'y$ gives $g^{-1}g'\in G_x$ .

(b) An isomorphism preserves stabilizers, so the classes agree. Conversely, if they agree, some $y\in Y$ has $G_y=H$ , and (a) gives an isomorphism. The object $G/H$ has class $[H]$ , so the assignment is onto.

(c) By (a) with $Y=X$ , the automorphisms correspond to the points of $X_H$ , and a point $nx$ has stabilizer $nHn^{-1}$ , so $X_H=N_G(H)x$ . Writing $\phi_n$ for the automorphism with $\phi_n(x)=nx$ , one has $\phi_n\phi_m=\phi_{mn}$ , so $n\mapsto\phi_{n^{-1}}$ is a homomorphism $N_G(H)\to\Aut_G(X)$ , onto by (a), and $\phi_n$ is the identity exactly when $n\in H$ .

(d) If $f,f'$ are isomorphisms, then $f^{-1}f'\in\Aut_G(X)$ and $f'=f\circ(f^{-1}f')$ , and $f\circ a=f\circ b$ forces $a=b$ . The same argument applies to postcomposition, and the count follows from (c).

Remark(The classification as a groupoid)

The groupoid whose vertices are the objects of $G$ and whose arrows are the $G$ -isomorphisms is equivalent to the disjoint union, over the conjugacy classes $[H]$ of subgroups, of the groups $N_G(H)/H$ , each regarded as a groupoid with one vertex. Thus an entry of an atlas of objects is a conjugacy class of subgroups, and the residual freedom in matching its incarnations is the group $N_G(H)/H$ .

Parts (b) and (c) of the principle are classical; in the language of permutation groups, $\Aut_G(X)$ is the centralizer of $G$ in $\operatorname{Sym}(X)$ .

Examplecomputed

The group $G=\PSL(2,7)$ of order 168 has exactly 179 subgroups, in fifteen conjugacy classes, in agreement with Dickson’s classification. So it has exactly fifteen objects, of sizes

168,\ 84,\ 56,\ 42,\ 42,\ 42,\ 28,\ 24,\ 21,\ 14,\ 14,\ 8,\ 7,\ 7,\ 1,

with stabilizers 1, $C_2$ , $C_3$ , $C_4$ , $V_4^a$ , $V_4^b$ , $S_3$ , $C_7$ , $D_8$ , $A_4^a$ , $A_4^b$ , $7{:}3$ , $S_4^a$ , $S_4^b$ and $G$ . The labels $a$ and $b$ are fixed by a marking: $S_4^a$ is the class of the stabilizers of the points of the Fano plane, $S_4^b$ that of its lines, and $V_4^a$ , $A_4^a$ lie in a member of $S_4^a$ as its normal Klein four-group and its alternating group.