The geometry of the groups PSL ( 2 , q ) Julie De Saedeleer - PowerPoint PPT Presentation

Motivation Definitions Main steps of the classification Self-normalizing and Borel-self-normalizing Locally s -arc-transitive graphs The geometry of the groups PSL ( 2 , q ) Julie De Saedeleer Université Libre de Bruxelles October 21th, 2011

Motivation Definitions Main steps of the classification Self-normalizing and Borel-self-normalizing Locally s -arc-transitive graphs Classification of the finite simple groups Context : Classification of the finite simple groups. 18 infinite families. 26 sporadic groups. Question left open : To achieve a unified geometric interpretation of all finite simple groups (Buekenhout). Encouragement in this direction : Theory of Buildings by J. Tits. Applies to 17 of the 18 infinite families leaving aside the Alt ( n ) and the 26 sporadic groups.

Motivation Definitions Main steps of the classification Self-normalizing and Borel-self-normalizing Locally s -arc-transitive graphs Classification of the finite simple groups Context : Classification of the finite simple groups. 18 infinite families. 26 sporadic groups. Question left open : To achieve a unified geometric interpretation of all finite simple groups (Buekenhout). Encouragement in this direction : Theory of Buildings by J. Tits. Applies to 17 of the 18 infinite families leaving aside the Alt ( n ) and the 26 sporadic groups.

Motivation Definitions Main steps of the classification Self-normalizing and Borel-self-normalizing Locally s -arc-transitive graphs Classification of the finite simple groups Context : Classification of the finite simple groups. 18 infinite families. 26 sporadic groups. Question left open : To achieve a unified geometric interpretation of all finite simple groups (Buekenhout). Encouragement in this direction : Theory of Buildings by J. Tits. Applies to 17 of the 18 infinite families leaving aside the Alt ( n ) and the 26 sporadic groups.

Motivation Definitions Main steps of the classification Self-normalizing and Borel-self-normalizing Locally s -arc-transitive graphs Classification of the finite simple groups Context : Classification of the finite simple groups. 18 infinite families. 26 sporadic groups. Question left open : To achieve a unified geometric interpretation of all finite simple groups (Buekenhout). Encouragement in this direction : Theory of Buildings by J. Tits. Applies to 17 of the 18 infinite families leaving aside the Alt ( n ) and the 26 sporadic groups.

Motivation Definitions Main steps of the classification Self-normalizing and Borel-self-normalizing Locally s -arc-transitive graphs Classification of the finite simple groups Context : Classification of the finite simple groups. 18 infinite families. 26 sporadic groups. Question left open : To achieve a unified geometric interpretation of all finite simple groups (Buekenhout). Encouragement in this direction : Theory of Buildings by J. Tits. Applies to 17 of the 18 infinite families leaving aside the Alt ( n ) and the 26 sporadic groups.

Motivation Definitions Main steps of the classification Self-normalizing and Borel-self-normalizing Locally s -arc-transitive graphs Classification of the finite simple groups Context : Classification of the finite simple groups. 18 infinite families. 26 sporadic groups. Question left open : To achieve a unified geometric interpretation of all finite simple groups (Buekenhout). Encouragement in this direction : Theory of Buildings by J. Tits. Applies to 17 of the 18 infinite families leaving aside the Alt ( n ) and the 26 sporadic groups.

Motivation Definitions Main steps of the classification Self-normalizing and Borel-self-normalizing Locally s -arc-transitive graphs Two main traces have been developed in incidence geometry Classify geometries over a given diagram 1 Given a group G , classify all incidence geometries of this 2 group and find a good set of axioms to impose on them. This subject is known as Coset geometry

Motivation Definitions Main steps of the classification Self-normalizing and Borel-self-normalizing Locally s -arc-transitive graphs Two main traces have been developed in incidence geometry Classify geometries over a given diagram 1 Given a group G , classify all incidence geometries of this 2 group and find a good set of axioms to impose on them. This subject is known as Coset geometry

Motivation Definitions Main steps of the classification Self-normalizing and Borel-self-normalizing Locally s -arc-transitive graphs Sample of known results Sample of known results, theorical and experimental Every Alt ( n ) and Sym ( n ) for n ≤ 8 (Cara) Sporadic groups (Buekenhout, Dehon, Gottchalk, Leemans, Miller): M 11 , M 12 , M 22 , M 23 , M 24 , J 1 , J 2 , J 3 , HS , Mcl O’Nan (partial results) Sz (Leemans) Every PSL ( 2 , q ) for q ≤ 19 (Cara, Dehon, Leemans, Vanmeerbeek)

Motivation Definitions Main steps of the classification Self-normalizing and Borel-self-normalizing Locally s -arc-transitive graphs On the way to classify all geometries of PSL ( 2 , q ) Idea Classify all coset geometries for every PSL ( 2 , q ) ( q prime-power).

Motivation Definitions Main steps of the classification Self-normalizing and Borel-self-normalizing Locally s -arc-transitive graphs On the way to classify all geometries of PSL ( 2 , q ) Idea Classify all coset geometries for every PSL ( 2 , q ) ( q prime-power). Classification of all coset geometries of rank two on which some group PSL ( 2 , q ) , q a prime power, acts flag-transitively. Classification under additional conditions, to be explained.

Motivation Definitions Main steps of the classification Self-normalizing and Borel-self-normalizing Locally s -arc-transitive graphs Incidence geometry of rank two Geometry of rank two A geometry Γ is a four-tuple ( X , ∗ , t , I ) where X is a set whose elements are called the elements of Γ ; 1 I is the set { 0 , 1 } whose elements are called the types of Γ ; 2 t : X → I is a mapping from X onto I; 3 ∗ is a symmetric and reflexive relation on X × X such that 4 no two distinct elements of the same type are incident.

Motivation Definitions Main steps of the classification Self-normalizing and Borel-self-normalizing Locally s -arc-transitive graphs Incidence geometry of rank two Geometry of rank two A geometry Γ is a four-tuple ( X , ∗ , t , I ) where X is a set whose elements are called the elements of Γ ; 1 I is the set { 0 , 1 } whose elements are called the types of Γ ; 2 t : X → I is a mapping from X onto I; 3 ∗ is a symmetric and reflexive relation on X × X such that 4 no two distinct elements of the same type are incident. Every element of a given type is incident to at least one element of the other type.

Motivation Definitions Main steps of the classification Self-normalizing and Borel-self-normalizing Locally s -arc-transitive graphs Incidence geometry of rank two Geometry of rank two A geometry Γ is a four-tuple ( X , ∗ , t , I ) where X is a set whose elements are called the elements of Γ ; 1 I is the set { 0 , 1 } whose elements are called the types of Γ ; 2 t : X → I is a mapping from X onto I; 3 ∗ is a symmetric and reflexive relation on X × X such that 4 no two distinct elements of the same type are incident. Every element of a given type is incident to at least one element of the other type. Flag In a geometry, a flag F is a set of pairwise incident elements.

Motivation Definitions Main steps of the classification Self-normalizing and Borel-self-normalizing Locally s -arc-transitive graphs Coset Geometry: Definition (due to Tits) Let I = { 0 , 1 } be the type set; let G be a group with two distinct subgroups ( G i ) i ∈ I .

Motivation Definitions Main steps of the classification Self-normalizing and Borel-self-normalizing Locally s -arc-transitive graphs Coset Geometry: Definition (due to Tits) Let I = { 0 , 1 } be the type set; let G be a group with two distinct subgroups ( G i ) i ∈ I . We require: G = < G 0 , G 1 > ; 1 G 0 ∩ G 1 is a proper subgroup of G 0 and of G 1 . 2

Motivation Definitions Main steps of the classification Self-normalizing and Borel-self-normalizing Locally s -arc-transitive graphs Coset Geometry: Definition (due to Tits) Let I = { 0 , 1 } be the type set; let G be a group with two distinct subgroups ( G i ) i ∈ I . We require: G 0 ∩ G 1 is a proper subgroup of G 0 and of G 1 . 1 G = < G 0 , G 1 > ; 2 Construction of a Coset geometry for ( G , { G 0 , G 1 } ) We construct a geometry Γ = Γ( G , ( G i ) i ∈ I ) = ( X , t , ∗ , I ) as follows The set of elements is X = { gG i | g ∈ G , G i ∈ ( G i ) i ∈ I } . 1 We define an incidence relation ∗ on X × X by 2 � gG i ∗ hG j ⇔ gG i hG j � = ∅ The type function on Γ is defined by t ( gG i ) = i 3

Motivation Definitions Main steps of the classification Self-normalizing and Borel-self-normalizing Locally s -arc-transitive graphs Coset Geometry: Definition (due to Tits) Let I = { 0 , 1 } be the type set; let G be a group with two distinct subgroups ( G i ) i ∈ I . We require: G 0 ∩ G 1 is a proper subgroup of G 0 and of G 1 . 1 G = < G 0 , G 1 > ; 2 Construction of a Coset geometry for ( G , { G 0 , G 1 } ) We construct a geometry Γ = Γ( G , ( G i ) i ∈ I ) = ( X , t , ∗ , I ) as follows The set of elements is X = { gG i | g ∈ G , G i ∈ ( G i ) i ∈ I } . 1 We define an incidence relation ∗ on X × X by 2 � gG i ∗ hG j ⇔ gG i hG j � = ∅ The type function on Γ is defined by t ( gG i ) = i 3