life

Floor 4, Les doubles vies · introduced in Chapter 6, Quatre groupes à double vie

In which classical geometry does a group live?

An isomorphism of a group onto a member of the families PSL(n,q) or A_m: a marking whose target brings a classical geometry with it.

Plate 4.1The line life inside the plane life: the eight points of

\Proj^1(\F_7)

are the eight Singer subgroups of the Fano plane, matched by one of 336 bijections.

Definition(Life, double life, dictionary)

The families are the groups $\PSL(n,q)$ , with $n\ge2$ and $(n,q)\neq(2,2),(2,3)$ , each acting on its projective space $\mathrm{PG}(n-1,q)$ , and the alternating groups $A_m$ , $m\ge5$ , each acting on $m$ letters. A life of a finite group $G$ is an isomorphism $\mu\colon G\to\Gamma$ onto a member $\Gamma$ of one of the two families: a marking whose target is a member of a family.

The natural sets of the life are the transitive $\Gamma$ -sets that the geometry of $\Gamma$ supplies directly: points, subspaces and flags of $\mathrm{PG}(n-1,q)$ ; subsets and partitions of the $m$ letters. Through $\mu$ they are objects of $G$ .

Proposition(From the plane to the line) computed

The group $\GL(3,2)$ has eight Sylow 7-subgroups and acts on them by conjugation. Exactly 336 bijections from this set onto $\Proj^1(\F_7)$ carry the resulting permutation group onto $\PSL(2,7)$ acting by Möbius transformations, and each defines an isomorphism $\GL(3,2)\to\PSL(2,7)$ .

A Sylow 7-subgroup of $\GL(3,2)$ is generated by a Singer cycle, a linear map of order 7 permuting the seven points of the plane cyclically. So the projective line over $\F_7$ is, in the plane’s own terms, the set of its eight Singer subgroups.

Proof

By machine, testing all $8!$ bijections against two generators. The count agrees with the theorem on seams without markings: the stabilizer of a Sylow 7-subgroup is its normalizer, of order 21, self-normalizing and the only class of subgroups of its order, so the permutation isomorphisms number $|\Aut\PSL(2,7)|=336$ .

Example

At 168 the two lives are $\PSL(2,7)$ on $\Proj^1(\F_7)$ and $\PSL(3,2)$ on $\mathrm{PG}(2,2)$ . The object of size 8 is the points in the line life and the Singer subgroups, or cyclic orientations, in the plane life. In the other direction the points and the lines of the plane are Galois’s two seven-point actions of $\PSL(2,7)$ , exchanged by an outer automorphism, and in the line life they are two orbits of bisections of $\Proj^1(\F_7)$ .

Remark

To ask which lives lift to characteristic zero, the definition widens to every group of Lie type: a life of a simple group $S$ is then an isomorphism of $S$ with $X/Z(X)$ , where $X$ is a linear, unitary, symplectic or orthogonal group, or an exceptional group, with its natural module over a field of characteristic $p$ . Isomorphisms in one characteristic, such as $\Omega(5,q)\cong\mathrm{PSp}(4,q)$ , then give one group several lives with different natural modules. Chapter 18 answers the question.

Built from: markingobject
Builds: double lifecompletiontype law
Objects: the skythe seven pointsthe seven lines
In the Esquisse: 8La famille de Weyl9Immeubles et réseaux11Le revêtement double et le miroir12Où se rencontrent les deux parents16Une loi de réciprocité18Exceptionnel veut dire relevable19Une formule du produit
The volume’s word: sky
In the volume: XThe Finite Celestial SphereXVIIITwo Parents of the Sky