Saturated fusion systems over a Sylow p -subgroup of Sp 4 ( p n ) - PowerPoint PPT Presentation

Saturated fusion systems over a Sylow p -subgroup of Sp 4 ( p n ) Ellen Henke (Aberdeen) Joint work with Sergey Shpectorov (Birmingham) Saturated fusion systems over a Sylow p -subgroup of Sp 4 ( p n ) Ellen Henke (Aberdeen)

Fusion in groups Traditionally, the term fusion refers to conjugacy of p -elements and p -subgroups of a fixed (finite) group G . Saturated fusion systems over a Sylow p -subgroup of Sp 4 ( p n ) Ellen Henke (Aberdeen)

Fusion in groups Traditionally, the term fusion refers to conjugacy of p -elements and p -subgroups of a fixed (finite) group G . So we study the homomorphisms c g : P → Q defined by x �→ x g := g − 1 xg , where P , Q ≤ G are p -subgroups and g ∈ G with P g ≤ Q . Saturated fusion systems over a Sylow p -subgroup of Sp 4 ( p n ) Ellen Henke (Aberdeen)

Fusion in groups Traditionally, the term fusion refers to conjugacy of p -elements and p -subgroups of a fixed (finite) group G . So we study the homomorphisms c g : P → Q defined by x �→ x g := g − 1 xg , where P , Q ≤ G are p -subgroups and g ∈ G with P g ≤ Q . If G is a finite group with Sylow p -subgroup S , the information about fusion in G is basically encoded in the following category: Saturated fusion systems over a Sylow p -subgroup of Sp 4 ( p n ) Ellen Henke (Aberdeen)

Fusion in groups Traditionally, the term fusion refers to conjugacy of p -elements and p -subgroups of a fixed (finite) group G . So we study the homomorphisms c g : P → Q defined by x �→ x g := g − 1 xg , where P , Q ≤ G are p -subgroups and g ∈ G with P g ≤ Q . If G is a finite group with Sylow p -subgroup S , the information about fusion in G is basically encoded in the following category: Definition. Let G be a group and S a subgroup of G . The fusion category F S ( G ) is the category whose objects are all subgroups of S and, for all subgroups P , Q ≤ S , Mor F S ( G ) ( P , Q ) := Hom G ( P , Q ) { c g : P → Q | g ∈ G with P g ≤ Q } . := Saturated fusion systems over a Sylow p -subgroup of Sp 4 ( p n ) Ellen Henke (Aberdeen)

Fusion Systems Definition. Let S be a finite p -group. A fusion system on S is a category F such that the objects are all subgroups of S and the following axioms hold for all P , Q ≤ S : Saturated fusion systems over a Sylow p -subgroup of Sp 4 ( p n ) Ellen Henke (Aberdeen)

Fusion Systems Definition. Let S be a finite p -group. A fusion system on S is a category F such that the objects are all subgroups of S and the following axioms hold for all P , Q ≤ S : Hom S ( P , Q ) ⊆ Mor F ( P , Q ) ⊆ Inj( P , Q ). Saturated fusion systems over a Sylow p -subgroup of Sp 4 ( p n ) Ellen Henke (Aberdeen)

Fusion Systems Definition. Let S be a finite p -group. A fusion system on S is a category F such that the objects are all subgroups of S and the following axioms hold for all P , Q ≤ S : Hom S ( P , Q ) ⊆ Mor F ( P , Q ) ⊆ Inj( P , Q ). (In particular, if P ≤ Q then the inclusion map P → Q is in Mor F ( P , Q ). Moreover, id P ∈ Mor F ( P , P ).) Saturated fusion systems over a Sylow p -subgroup of Sp 4 ( p n ) Ellen Henke (Aberdeen)

Fusion Systems Definition. Let S be a finite p -group. A fusion system on S is a category F such that the objects are all subgroups of S and the following axioms hold for all P , Q ≤ S : Hom S ( P , Q ) ⊆ Mor F ( P , Q ) ⊆ Inj( P , Q ). (In particular, if P ≤ Q then the inclusion map P → Q is in Mor F ( P , Q ). Moreover, id P ∈ Mor F ( P , P ).) Every ϕ ∈ Mor F ( P , Q ) is the composite of an F -isomorphism followed by an inclusion. Saturated fusion systems over a Sylow p -subgroup of Sp 4 ( p n ) Ellen Henke (Aberdeen)

Fusion Systems Definition. Let S be a finite p -group. A fusion system on S is a category F such that the objects are all subgroups of S and the following axioms hold for all P , Q ≤ S : Hom S ( P , Q ) ⊆ Mor F ( P , Q ) ⊆ Inj( P , Q ). (In particular, if P ≤ Q then the inclusion map P → Q is in Mor F ( P , Q ). Moreover, id P ∈ Mor F ( P , P ).) Every ϕ ∈ Mor F ( P , Q ) is the composite of an F -isomorphism followed by an inclusion. (That means ϕ : P → ϕ ( P ) and ϕ − 1 : ϕ ( P ) → P are morphisms in F .) Saturated fusion systems over a Sylow p -subgroup of Sp 4 ( p n ) Ellen Henke (Aberdeen)

Fusion Systems Definition. Let S be a finite p -group. A fusion system on S is a category F such that the objects are all subgroups of S and the following axioms hold for all P , Q ≤ S : Hom S ( P , Q ) ⊆ Mor F ( P , Q ) ⊆ Inj( P , Q ). (In particular, if P ≤ Q then the inclusion map P → Q is in Mor F ( P , Q ). Moreover, id P ∈ Mor F ( P , P ).) Every ϕ ∈ Mor F ( P , Q ) is the composite of an F -isomorphism followed by an inclusion. (That means ϕ : P → ϕ ( P ) and ϕ − 1 : ϕ ( P ) → P are morphisms in F .) Example. If G is a group with a finite p -subgroup S , then F S ( G ) is a fusion system on S . Saturated fusion systems over a Sylow p -subgroup of Sp 4 ( p n ) Ellen Henke (Aberdeen)

Fusion Systems Definition. Let S be a finite p -group. A fusion system on S is a category F such that the objects are all subgroups of S and the following axioms hold for all P , Q ≤ S : Hom S ( P , Q ) ⊆ Mor F ( P , Q ) ⊆ Inj( P , Q ). (In particular, if P ≤ Q then the inclusion map P → Q is in Mor F ( P , Q ). Moreover, id P ∈ Mor F ( P , P ).) Every ϕ ∈ Mor F ( P , Q ) is the composite of an F -isomorphism followed by an inclusion. (That means ϕ : P → ϕ ( P ) and ϕ − 1 : ϕ ( P ) → P are morphisms in F .) Example. If G is a group with a finite p -subgroup S , then F S ( G ) is a fusion system on S . If G is finite and S ∈ Syl p ( G ) then F S ( G ) has “nice properties”. Saturated fusion systems over a Sylow p -subgroup of Sp 4 ( p n ) Ellen Henke (Aberdeen)

Notation From now on let F be a fusion system on a finite p -group S . Saturated fusion systems over a Sylow p -subgroup of Sp 4 ( p n ) Ellen Henke (Aberdeen)

Notation From now on let F be a fusion system on a finite p -group S . For P , Q ≤ S set Hom F ( P , Q ) := Mor F ( P , Q ) , Saturated fusion systems over a Sylow p -subgroup of Sp 4 ( p n ) Ellen Henke (Aberdeen)

Notation From now on let F be a fusion system on a finite p -group S . For P , Q ≤ S set Hom F ( P , Q ) := Mor F ( P , Q ) , Aut F ( P ) := Hom F ( P , P ) , Saturated fusion systems over a Sylow p -subgroup of Sp 4 ( p n ) Ellen Henke (Aberdeen)

Notation From now on let F be a fusion system on a finite p -group S . For P , Q ≤ S set Hom F ( P , Q ) := Mor F ( P , Q ) , Aut F ( P ) := Hom F ( P , P ) , { ϕ ∈ Hom F ( P , Q ) | ϕ ( P ) = Q } . Iso F ( P , Q ) := Saturated fusion systems over a Sylow p -subgroup of Sp 4 ( p n ) Ellen Henke (Aberdeen)

Notation From now on let F be a fusion system on a finite p -group S . For P , Q ≤ S set Hom F ( P , Q ) := Mor F ( P , Q ) , Aut F ( P ) := Hom F ( P , P ) , { ϕ ∈ Hom F ( P , Q ) | ϕ ( P ) = Q } . Iso F ( P , Q ) := Set P F := { Q ≤ S : Iso F ( P , Q ) � = ∅} . The subgroups of S in P F are called F -conjugate to P . Saturated fusion systems over a Sylow p -subgroup of Sp 4 ( p n ) Ellen Henke (Aberdeen)

Saturation “Definition.” The fusion system F is called saturated if, for any P ≤ S , there exists a subgroup Q ∈ P F , such that the following properties hold: Saturated fusion systems over a Sylow p -subgroup of Sp 4 ( p n ) Ellen Henke (Aberdeen)

Saturation “Definition.” The fusion system F is called saturated if, for any P ≤ S , there exists a subgroup Q ∈ P F , such that the following properties hold: (I) Aut S ( Q ) ∈ Syl p (Aut F ( Q )). Saturated fusion systems over a Sylow p -subgroup of Sp 4 ( p n ) Ellen Henke (Aberdeen)

Saturation “Definition.” The fusion system F is called saturated if, for any P ≤ S , there exists a subgroup Q ∈ P F , such that the following properties hold: (I) Aut S ( Q ) ∈ Syl p (Aut F ( Q )). (II) Any F -morphism with image Q can be extended. More precisely, if ϕ ∈ Iso F ( R , Q ), then for a certain subgroup N ϕ ≤ N S ( R ), there exists ˆ ϕ ∈ Mor F ( N ϕ , S ) with ˆ ϕ | R = ϕ . Saturated fusion systems over a Sylow p -subgroup of Sp 4 ( p n ) Ellen Henke (Aberdeen)

Saturation “Definition.” The fusion system F is called saturated if, for any P ≤ S , there exists a subgroup Q ∈ P F , such that the following properties hold: (I) Aut S ( Q ) ∈ Syl p (Aut F ( Q )). (II) Any F -morphism with image Q can be extended. More precisely, if ϕ ∈ Iso F ( R , Q ), then for a certain subgroup N ϕ ≤ N S ( R ), there exists ˆ ϕ ∈ Mor F ( N ϕ , S ) with ˆ ϕ | R = ϕ . If F is saturated, then (I) and (II) hold for any fully normalized subgroup Q , i.e. for any subgroup Q such that | N S ( Q ) | ≥ | N S ( Q ∗ ) | for any Q ∗ ∈ Q F . Saturated fusion systems over a Sylow p -subgroup of Sp 4 ( p n ) Ellen Henke (Aberdeen)

Realizing fusion systems Examples: The fusion category F S ( G ) of a finite group G with Sylow p -subgroup S is saturated. Saturated fusion systems over a Sylow p -subgroup of Sp 4 ( p n ) Ellen Henke (Aberdeen)

Realizing fusion systems Examples: The fusion category F S ( G ) of a finite group G with Sylow p -subgroup S is saturated. Every p -block of a finite group leads to a saturated fusion system on the defect group. Saturated fusion systems over a Sylow p -subgroup of Sp 4 ( p n ) Ellen Henke (Aberdeen)