rigid object

Floor 2, Les sutures · introduced in Chapter 1, Un objet, plusieurs noms

When is the seam between two incarnations forced?

An object with no automorphism but the identity; equivalently its stabilizers are self-normalizing, and then every seam is unique.

Plate 2.3The six rigid objects of the group of order 168, in gold; beside each of the other nine, its automorphism group

N_G(H)/H

Definition(Rigid object)

An object is rigid if its only automorphism is the identity. The word is used in its combinatorial sense, a structure with no nontrivial automorphism.

Corollary(Rigidity criterion)

Let $X$ be an object with stabilizer class $[H]$ . The following are equivalent: (i) $H$ is self-normalizing; (ii) $X$ is rigid; (iii) every incarnation of $X$ has exactly one alignment; (iv) between any two incarnations of $X$ there is exactly one seam.

Proof

(i) and (ii) are equivalent because $\Aut_G(X)\cong N_G(H)/H$ . The alignments of an incarnation $Y$ form $\Iso_G(X,Y)$ and the seams between $Y$ and $Y'$ form $\Iso_G(Y,Y')$ ; both are torsors for $\Aut_G(X)$ , so both have $|\Aut_G(X)|$ elements.

Corollary(Rigid and non-rigid objects) computed

Exactly six of the fifteen objects of $\PSL(2,7)$ are rigid: 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. 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 $C_2$ .

Example

For a Sylow 3-subgroup $P$ of $\PSL(2,7)$ the normalizer $N_G(P)$ is an $S_3$ , and $P$ is its only subgroup of order 3, so anything normalizing $N_G(P)$ normalizes $P$ : $N_G(N_G(P))=N_G(P)$ . So the object of size 28 is rigid. Its class is the only class of subgroups of order 6, so it is also fixed by every automorphism of $G$ , and between any two of its incarnations there is exactly one seam, whatever the markings.