Non-Existence of Sharply 2-Transitive Sets of Permutations in Sp ( 2 - - PowerPoint PPT Presentation

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Non-Existence of Sharply 2-Transitive Sets of Permutations in Sp ( 2 - - PowerPoint PPT Presentation

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Non-Existence of Sharply 2-Transitive Sets of Permutations in Sp ( 2 d , 2 ) Dominik Barth University of Wrzburg May 14, 2016 Groups and Topological Groups 2016 Dominik Barth Sharply Transitive Sets Definitions Definitions Let be a

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Non-Existence of Sharply 2-Transitive Sets of Permutations in Sp(2d, 2)

Dominik Barth University of Würzburg May 14, 2016

Groups and Topological Groups 2016 Dominik Barth

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Sharply Transitive Sets — Definitions

Definitions

Let Ω be a finite set and S ⊆ Sym Ω.

Groups and Topological Groups 2016 Dominik Barth

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SLIDE 3

Sharply Transitive Sets — Definitions

Definitions

Let Ω be a finite set and S ⊆ Sym Ω. S is sharply transitive, if for all α, β ∈ Ω there is a unique g ∈ S with αg = β.

Groups and Topological Groups 2016 Dominik Barth

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Sharply Transitive Sets — Definitions

Definitions

Let Ω be a finite set and S ⊆ Sym Ω. S is sharply transitive, if for all α, β ∈ Ω there is a unique g ∈ S with αg = β. S is sharply 2-transitive, if S is sharply transitive on {(ω1, ω2) ∈ Ω2 | ω1 = ω2}.

Groups and Topological Groups 2016 Dominik Barth

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SLIDE 5

Sharply Transitive Sets — Definitions

Definitions

Let Ω be a finite set and S ⊆ Sym Ω. S is sharply transitive, if for all α, β ∈ Ω there is a unique g ∈ S with αg = β. S is sharply 2-transitive, if S is sharply transitive on {(ω1, ω2) ∈ Ω2 | ω1 = ω2}.

Observation (Witt)

Sharply 2-transitive subsets of Sn correspond to projective planes of order n.

Groups and Topological Groups 2016 Dominik Barth

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SLIDE 6

Sharply Transitive Sets — Definitions

Definitions

Let Ω be a finite set and S ⊆ Sym Ω. S is sharply transitive, if for all α, β ∈ Ω there is a unique g ∈ S with αg = β. S is sharply 2-transitive, if S is sharply transitive on {(ω1, ω2) ∈ Ω2 | ω1 = ω2}.

Observation (Witt)

Sharply 2-transitive subsets of Sn correspond to projective planes of order n.

Problem (Hard)

Show the non-existence of sharply 2-transitive sets in Sn.

Groups and Topological Groups 2016 Dominik Barth

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Sharply Transitive Sets — Definitions

Definitions

Let Ω be a finite set and S ⊆ Sym Ω. S is sharply transitive, if for all α, β ∈ Ω there is a unique g ∈ S with αg = β. S is sharply 2-transitive, if S is sharply transitive on {(ω1, ω2) ∈ Ω2 | ω1 = ω2}.

Observation (Witt)

Sharply 2-transitive subsets of Sn correspond to projective planes of order n.

Problem (Easier)

Show the non-existence of sharply 2-transitive sets in subgroups of Sn.

Groups and Topological Groups 2016 Dominik Barth

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Sharply Transitive Sets — Definitions

Definitions

Let Ω be a finite set and S ⊆ Sym Ω. S is sharply transitive, if for all α, β ∈ Ω there is a unique g ∈ S with αg = β. S is sharply 2-transitive, if S is sharply transitive on {(ω1, ω2) ∈ Ω2 | ω1 = ω2}.

Observation (Witt)

Sharply 2-transitive subsets of Sn correspond to projective planes of order n.

Problem (Lorimer, 1970s)

Show the non-existence of sharply 2-transitive sets in 2-transitive subgroups of Sn.

Groups and Topological Groups 2016 Dominik Barth

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Sharply Transitive Sets — Results

Problem (Lorimer, 1970s)

Show the non-existence of sharply 2-transitive sets in 2-transitive subgroups of Sn.

Groups and Topological Groups 2016 Dominik Barth

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Sharply Transitive Sets — Results

Problem (Lorimer, 1970s)

Show the non-existence of sharply 2-transitive sets in 2-transitive subgroups of Sn.

Theorem (Lorimer, O’Nan, Grundhöfer, Müller, Nagy)

Suppose G ≤ Sn contains a sharply 2-transitive subset. Then:

Groups and Topological Groups 2016 Dominik Barth

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Sharply Transitive Sets — Results

Problem (Lorimer, 1970s)

Show the non-existence of sharply 2-transitive sets in 2-transitive subgroups of Sn.

Theorem (Lorimer, O’Nan, Grundhöfer, Müller, Nagy)

Suppose G ≤ Sn contains a sharply 2-transitive subset. Then: G ≤ AGLe(Fp), n = pe, or

Groups and Topological Groups 2016 Dominik Barth

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Sharply Transitive Sets — Results

Problem (Lorimer, 1970s)

Show the non-existence of sharply 2-transitive sets in 2-transitive subgroups of Sn.

Theorem (Lorimer, O’Nan, Grundhöfer, Müller, Nagy)

Suppose G ≤ Sn contains a sharply 2-transitive subset. Then: G ≤ AGLe(Fp), n = pe, or G = An, n ≡ 0, 1 mod 4, or

Groups and Topological Groups 2016 Dominik Barth

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Sharply Transitive Sets — Results

Problem (Lorimer, 1970s)

Show the non-existence of sharply 2-transitive sets in 2-transitive subgroups of Sn.

Theorem (Lorimer, O’Nan, Grundhöfer, Müller, Nagy)

Suppose G ≤ Sn contains a sharply 2-transitive subset. Then: G ≤ AGLe(Fp), n = pe, or G = An, n ≡ 0, 1 mod 4, or G = Sn, or

Groups and Topological Groups 2016 Dominik Barth

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SLIDE 14

Sharply Transitive Sets — Results

Problem (Lorimer, 1970s)

Show the non-existence of sharply 2-transitive sets in 2-transitive subgroups of Sn.

Theorem (Lorimer, O’Nan, Grundhöfer, Müller, Nagy)

Suppose G ≤ Sn contains a sharply 2-transitive subset. Then: G ≤ AGLe(Fp), n = pe, or G = An, n ≡ 0, 1 mod 4, or G = Sn, or G = M24.

Groups and Topological Groups 2016 Dominik Barth

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Sharply Transitive Sets — Methods

What methods can be used to prove the non-existence of sharply transitive sets?

Groups and Topological Groups 2016 Dominik Barth

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Sharply Transitive Sets — Methods

What methods can be used to prove the non-existence of sharply transitive sets? “old”: (Modular) character theory (O’Nan 1984, Grundhöfer-Müller 2009)

Groups and Topological Groups 2016 Dominik Barth

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Sharply Transitive Sets — Methods

What methods can be used to prove the non-existence of sharply transitive sets? “old”: (Modular) character theory (O’Nan 1984, Grundhöfer-Müller 2009) “new”: Contradicting subsets (Müller-Nagy 2011)

Groups and Topological Groups 2016 Dominik Barth

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Sharply Transitive Sets — Methods

What methods can be used to prove the non-existence of sharply transitive sets? “old”: (Modular) character theory (O’Nan 1984, Grundhöfer-Müller 2009) “new”: Contradicting subsets (Müller-Nagy 2011) Almost all “old” results were reproved using contradicting subsets.

Groups and Topological Groups 2016 Dominik Barth

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Sharply Transitive Sets — Methods

What methods can be used to prove the non-existence of sharply transitive sets? “old”: (Modular) character theory (O’Nan 1984, Grundhöfer-Müller 2009) “new”: Contradicting subsets (Müller-Nagy 2011) Almost all “old” results were reproved using contradicting subsets.

Exception

Non-existence of sharply 2-transitive sets in Sp(2d, 2) of degree 22d−1 ± 2d−1. (proved by Grundhöfer-Müller)

Groups and Topological Groups 2016 Dominik Barth

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Sharply Transitive Sets — Methods

What methods can be used to prove the non-existence of sharply transitive sets? “old”: (Modular) character theory (O’Nan 1984, Grundhöfer-Müller 2009) “new”: Contradicting subsets (Müller-Nagy 2011) Almost all “old” results were reproved using contradicting subsets.

Exception

Non-existence of sharply 2-transitive sets in Sp(2d, 2) of degree 22d−1 ± 2d−1. (proved by Grundhöfer-Müller)

In this talk

Contradicting subsets for all Sp(2d, 2), d ≥ 4

Groups and Topological Groups 2016 Dominik Barth

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SLIDE 21

Contradicting Subsets

Lemma (Müller, Nagy)

Let S ⊆ Sym Ω be sharply transitive and B, C ⊆ Ω arbitrary. Then

Groups and Topological Groups 2016 Dominik Barth

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SLIDE 22

Contradicting Subsets

Lemma (Müller, Nagy)

Let S ⊆ Sym Ω be sharply transitive and B, C ⊆ Ω arbitrary. Then |B||C| =

  • g∈S

|B ∩ Cg|.

Groups and Topological Groups 2016 Dominik Barth

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SLIDE 23

Contradicting Subsets

Lemma (Müller, Nagy)

Let S ⊆ Sym Ω be sharply transitive and B, C ⊆ Ω arbitrary. Then |B||C| =

  • g∈S

|B ∩ Cg|.

Proof.

Double counting of the set {(b, c, g) ∈ B × C × S | cg = b}.

Groups and Topological Groups 2016 Dominik Barth

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SLIDE 24

Contradicting Subsets

Lemma (Müller, Nagy)

Let S ⊆ Sym Ω be sharply transitive and B, C ⊆ Ω arbitrary. Then |B||C| =

  • g∈S

|B ∩ Cg|.

Proof.

Double counting of the set {(b, c, g) ∈ B × C × S | cg = b}.

Definition

Subsets B, C ⊆ Ω are contradicting subsets (modulo k) for G ≤ Sym Ω, if:

Groups and Topological Groups 2016 Dominik Barth

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SLIDE 25

Contradicting Subsets

Lemma (Müller, Nagy)

Let S ⊆ Sym Ω be sharply transitive and B, C ⊆ Ω arbitrary. Then |B||C| =

  • g∈S

|B ∩ Cg|.

Proof.

Double counting of the set {(b, c, g) ∈ B × C × S | cg = b}.

Definition

Subsets B, C ⊆ Ω are contradicting subsets (modulo k) for G ≤ Sym Ω, if: k divides |B ∩ Cg| for all g ∈ G, but

Groups and Topological Groups 2016 Dominik Barth

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SLIDE 26

Contradicting Subsets

Lemma (Müller, Nagy)

Let S ⊆ Sym Ω be sharply transitive and B, C ⊆ Ω arbitrary. Then |B||C| =

  • g∈S

|B ∩ Cg|.

Proof.

Double counting of the set {(b, c, g) ∈ B × C × S | cg = b}.

Definition

Subsets B, C ⊆ Ω are contradicting subsets (modulo k) for G ≤ Sym Ω, if: k divides |B ∩ Cg| for all g ∈ G, but k does not divide |B||C|.

Groups and Topological Groups 2016 Dominik Barth

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SLIDE 27

Contradicting Subsets

Lemma (Müller, Nagy)

Let S ⊆ Sym Ω be sharply transitive and B, C ⊆ Ω arbitrary. Then |B||C| =

  • g∈S

|B ∩ Cg|.

Proof.

Double counting of the set {(b, c, g) ∈ B × C × S | cg = b}.

Definition

Subsets B, C ⊆ Ω are contradicting subsets (modulo k) for G ≤ Sym Ω, if: k divides |B ∩ Cg| for all g ∈ G, but k does not divide |B||C|.

  • =

⇒ ∄ sharply transitive S ⊆ G

Groups and Topological Groups 2016 Dominik Barth

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The 2-Transitive Actions of Sp(2d, 2)

Let (V = F2d

2 , ϕ) be a symplectic space,

i.e. ϕ : V × V → F2 is alternating, bilinear, non-degenerate.

Groups and Topological Groups 2016 Dominik Barth

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The 2-Transitive Actions of Sp(2d, 2)

Let (V = F2d

2 , ϕ) be a symplectic space,

i.e. ϕ : V × V → F2 is alternating, bilinear, non-degenerate. Quadratic forms that polarize to ϕ : Ω := {θ : V → F2 | θ(v + w) = θ(v) + θ(w) + ϕ(v, w) ∀v, w ∈ V}

Groups and Topological Groups 2016 Dominik Barth

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The 2-Transitive Actions of Sp(2d, 2)

Let (V = F2d

2 , ϕ) be a symplectic space,

i.e. ϕ : V × V → F2 is alternating, bilinear, non-degenerate. Quadratic forms that polarize to ϕ : Ω := {θ : V → F2 | θ(v + w) = θ(v) + θ(w) + ϕ(v, w) ∀v, w ∈ V} Singular vectors: V 0 := {v ∈ V | θ(v) = 0, v = 0}

Groups and Topological Groups 2016 Dominik Barth

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The 2-Transitive Actions of Sp(2d, 2)

Let (V = F2d

2 , ϕ) be a symplectic space,

i.e. ϕ : V × V → F2 is alternating, bilinear, non-degenerate. Quadratic forms that polarize to ϕ : Ω := {θ : V → F2 | θ(v + w) = θ(v) + θ(w) + ϕ(v, w) ∀v, w ∈ V} Singular vectors: V 0 := {v ∈ V | θ(v) = 0, v = 0}

Observations

Groups and Topological Groups 2016 Dominik Barth

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SLIDE 32

The 2-Transitive Actions of Sp(2d, 2)

Let (V = F2d

2 , ϕ) be a symplectic space,

i.e. ϕ : V × V → F2 is alternating, bilinear, non-degenerate. Quadratic forms that polarize to ϕ : Ω := {θ : V → F2 | θ(v + w) = θ(v) + θ(w) + ϕ(v, w) ∀v, w ∈ V} Singular vectors: V 0 := {v ∈ V | θ(v) = 0, v = 0}

Observations

Sp(V, ϕ) acts on Ω via θg(v) = θ(vg−1).

Groups and Topological Groups 2016 Dominik Barth

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The 2-Transitive Actions of Sp(2d, 2)

Let (V = F2d

2 , ϕ) be a symplectic space,

i.e. ϕ : V × V → F2 is alternating, bilinear, non-degenerate. Quadratic forms that polarize to ϕ : Ω := {θ : V → F2 | θ(v + w) = θ(v) + θ(w) + ϕ(v, w) ∀v, w ∈ V} Singular vectors: V 0 := {v ∈ V | θ(v) = 0, v = 0}

Observations

Sp(V, ϕ) acts on Ω via θg(v) = θ(vg−1). 2-transitive on both orbits Ω+ (Witt index d) and Ω− (Witt index d − 1).

Groups and Topological Groups 2016 Dominik Barth

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SLIDE 34

The 2-Transitive Actions of Sp(2d, 2)

Let (V = F2d

2 , ϕ) be a symplectic space,

i.e. ϕ : V × V → F2 is alternating, bilinear, non-degenerate. Quadratic forms that polarize to ϕ : Ω := {θ : V → F2 | θ(v + w) = θ(v) + θ(w) + ϕ(v, w) ∀v, w ∈ V} Singular vectors: V 0 := {v ∈ V | θ(v) = 0, v = 0}

Observations

Sp(V, ϕ) acts on Ω via θg(v) = θ(vg−1). 2-transitive on both orbits Ω+ (Witt index d) and Ω− (Witt index d − 1). Fix θ ∈ Ω±. Actions of Sp(V, ϕ)θ = O(V, θ) on Ω± \ {θ} and V 0 are equivalent.

Groups and Topological Groups 2016 Dominik Barth

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SLIDE 35

Contradicting Subsets for O±(2d, 2), d ≥ 4

Goal

Show: No sharply 2-transitive sets in Sp(V, ϕ) on both Ω+ and Ω−.

Groups and Topological Groups 2016 Dominik Barth

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SLIDE 36

Contradicting Subsets for O±(2d, 2), d ≥ 4

Goal

Show: No sharply 2-transitive sets in Sp(V, ϕ) on both Ω+ and Ω−.

How to achieve it

Show: No sharply transitive sets in O(V, θ) = Sp(V, ϕ)θ on V 0.

Groups and Topological Groups 2016 Dominik Barth

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SLIDE 37

Contradicting Subsets for O±(2d, 2), d ≥ 4

Goal

Show: No sharply 2-transitive sets in Sp(V, ϕ) on both Ω+ and Ω−.

How to achieve it

Show: No sharply transitive sets in O(V, θ) = Sp(V, ϕ)θ on V 0. Using contradicting subsets!

Groups and Topological Groups 2016 Dominik Barth

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SLIDE 38

Contradicting Subsets for O±(2d, 2), d ≥ 4

Goal

Show: No sharply 2-transitive sets in Sp(V, ϕ) on both Ω+ and Ω−.

How to achieve it

Show: No sharply transitive sets in O(V, θ) = Sp(V, ϕ)θ on V 0. Using contradicting subsets!

Ingredients for the construction of contradicting subsets:

Groups and Topological Groups 2016 Dominik Barth

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SLIDE 39

Contradicting Subsets for O±(2d, 2), d ≥ 4

Goal

Show: No sharply 2-transitive sets in Sp(V, ϕ) on both Ω+ and Ω−.

How to achieve it

Show: No sharply transitive sets in O(V, θ) = Sp(V, ϕ)θ on V 0. Using contradicting subsets!

Ingredients for the construction of contradicting subsets:

V = F2d

2 , d ≥ 4, quadratic form θ with polar form ϕ

Groups and Topological Groups 2016 Dominik Barth

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SLIDE 40

Contradicting Subsets for O±(2d, 2), d ≥ 4

Goal

Show: No sharply 2-transitive sets in Sp(V, ϕ) on both Ω+ and Ω−.

How to achieve it

Show: No sharply transitive sets in O(V, θ) = Sp(V, ϕ)θ on V 0. Using contradicting subsets!

Ingredients for the construction of contradicting subsets:

V = F2d

2 , d ≥ 4, quadratic form θ with polar form ϕ

non-degenerate U ≤ V with 4 ≤ dim U ≤ dim V − 4

Groups and Topological Groups 2016 Dominik Barth

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SLIDE 41

Contradicting Subsets for O±(2d, 2), d ≥ 4

Goal

Show: No sharply 2-transitive sets in Sp(V, ϕ) on both Ω+ and Ω−.

How to achieve it

Show: No sharply transitive sets in O(V, θ) = Sp(V, ϕ)θ on V 0. Using contradicting subsets!

Ingredients for the construction of contradicting subsets:

V = F2d

2 , d ≥ 4, quadratic form θ with polar form ϕ

non-degenerate U ≤ V with 4 ≤ dim U ≤ dim V − 4 W := U⊥ = ⇒ V = U⊥W.

Groups and Topological Groups 2016 Dominik Barth

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SLIDE 42

Contradicting Subsets for O±(2d, 2), d ≥ 4

Goal

Show: No sharply 2-transitive sets in Sp(V, ϕ) on both Ω+ and Ω−.

How to achieve it

Show: No sharply transitive sets in O(V, θ) = Sp(V, ϕ)θ on V 0. Using contradicting subsets!

Ingredients for the construction of contradicting subsets:

V = F2d

2 , d ≥ 4, quadratic form θ with polar form ϕ

non-degenerate U ≤ V with 4 ≤ dim U ≤ dim V − 4 W := U⊥ = ⇒ V = U⊥W. a non-singular vector c ∈ V

Groups and Topological Groups 2016 Dominik Barth

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SLIDE 43

Contradicting Subsets for O±(2d, 2), d ≥ 4

Reminder: V = U⊥W dim U, W ≥ 4 θ(c) = 1

Theorem (B.)

Setting

Groups and Topological Groups 2016 Dominik Barth

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SLIDE 44

Contradicting Subsets for O±(2d, 2), d ≥ 4

Reminder: V = U⊥W dim U, W ≥ 4 θ(c) = 1

Theorem (B.)

Setting B := U1 + W 1 = {u + w | u ∈ U, w ∈ W, θ(u) = θ(w) = 1} ⊆ V 0

Groups and Topological Groups 2016 Dominik Barth

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SLIDE 45

Contradicting Subsets for O±(2d, 2), d ≥ 4

Reminder: V = U⊥W dim U, W ≥ 4 θ(c) = 1

Theorem (B.)

Setting B := U1 + W 1 = {u + w | u ∈ U, w ∈ W, θ(u) = θ(w) = 1} ⊆ V 0 C := V 0 ∩ c⊥ = {v ∈ V | θ(v) = ϕ(v, c) = 0} \ {0}

Groups and Topological Groups 2016 Dominik Barth

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SLIDE 46

Contradicting Subsets for O±(2d, 2), d ≥ 4

Reminder: V = U⊥W dim U, W ≥ 4 θ(c) = 1

Theorem (B.)

Setting B := U1 + W 1 = {u + w | u ∈ U, w ∈ W, θ(u) = θ(w) = 1} ⊆ V 0 C := V 0 ∩ c⊥ = {v ∈ V | θ(v) = ϕ(v, c) = 0} \ {0} we have:

Groups and Topological Groups 2016 Dominik Barth

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SLIDE 47

Contradicting Subsets for O±(2d, 2), d ≥ 4

Reminder: V = U⊥W dim U, W ≥ 4 θ(c) = 1

Theorem (B.)

Setting B := U1 + W 1 = {u + w | u ∈ U, w ∈ W, θ(u) = θ(w) = 1} ⊆ V 0 C := V 0 ∩ c⊥ = {v ∈ V | θ(v) = ϕ(v, c) = 0} \ {0} we have: 2d−1 divides |B ∩ Cg| for all g ∈ O±(2d, 2), but

Groups and Topological Groups 2016 Dominik Barth

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SLIDE 48

Contradicting Subsets for O±(2d, 2), d ≥ 4

Reminder: V = U⊥W dim U, W ≥ 4 θ(c) = 1

Theorem (B.)

Setting B := U1 + W 1 = {u + w | u ∈ U, w ∈ W, θ(u) = θ(w) = 1} ⊆ V 0 C := V 0 ∩ c⊥ = {v ∈ V | θ(v) = ϕ(v, c) = 0} \ {0} we have: 2d−1 divides |B ∩ Cg| for all g ∈ O±(2d, 2), but 2d−1 is no divisor of |B||C|.

Groups and Topological Groups 2016 Dominik Barth

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SLIDE 49

Contradicting Subsets for O±(2d, 2), d ≥ 4

Reminder: V = U⊥W dim U, W ≥ 4 θ(c) = 1

Theorem (B.)

Setting B := U1 + W 1 = {u + w | u ∈ U, w ∈ W, θ(u) = θ(w) = 1} ⊆ V 0 C := V 0 ∩ c⊥ = {v ∈ V | θ(v) = ϕ(v, c) = 0} \ {0} we have: 2d−1 divides |B ∩ Cg| for all g ∈ O±(2d, 2), but 2d−1 is no divisor of |B||C|. Thus B, C are contradicting subsets for O±(2d, 2) on V 0.

Groups and Topological Groups 2016 Dominik Barth

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SLIDE 50

Contradicting Subsets for O±(2d, 2), d ≥ 4

Reminder: V = U⊥W dim U, W ≥ 4 θ(c) = 1

Theorem (B.)

Setting B := U1 + W 1 = {u + w | u ∈ U, w ∈ W, θ(u) = θ(w) = 1} ⊆ V 0 C := V 0 ∩ c⊥ = {v ∈ V | θ(v) = ϕ(v, c) = 0} \ {0} we have: 2d−1 divides |B ∩ Cg| for all g ∈ O±(2d, 2), but 2d−1 is no divisor of |B||C|. Thus B, C are contradicting subsets for O±(2d, 2) on V 0.

Corollary (Grundhöfer, Müller)

The symplectic groups Sp(2d, 2), d ≥ 4, in their actions of degrees 22d−1 ± 2d−1, have no sharply 2-transitive subsets.

Groups and Topological Groups 2016 Dominik Barth

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SLIDE 51

Proof of the Theorem

Groups and Topological Groups 2016 Dominik Barth

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SLIDE 52

Proof of the Theorem

Combinatorical building blocks

Groups and Topological Groups 2016 Dominik Barth

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SLIDE 53

Proof of the Theorem

Combinatorical building blocks

Let V be a 2d-dimensional non-degenerate orthogonal space over F2. Then:

Groups and Topological Groups 2016 Dominik Barth

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SLIDE 54

Proof of the Theorem

Combinatorical building blocks

Let V be a 2d-dimensional non-degenerate orthogonal space over F2. Then: (i) |V 0| = 22d−1 ±V 2d−1 − 1 and |V 1| = 22d−1 ∓V 2d−1. (ii) For v ∈ V 1 we have |V 1 ∩ v⊥| = |(V 0 ∩ v⊥) ∪ {0}| = 22d−2. (iii) For v ∈ V 1 we have |V 1 \ v⊥| = 22d−2 ∓V 2d−1. (iv) For v ∈ V 0 we have |V 1 \ v⊥| = 22d−2. (v) For v ∈ V 0 we have |V 1 ∩ v⊥| = 22d−2 ∓V 2d−1.

Groups and Topological Groups 2016 Dominik Barth

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SLIDE 55

Proof of the Theorem

Combinatorical building blocks

Let V be a 2d-dimensional non-degenerate orthogonal space over F2. Then: (i) |V 0| = 22d−1 ±V 2d−1 − 1 and |V 1| = 22d−1 ∓V 2d−1. (ii) For v ∈ V 1 we have |V 1 ∩ v⊥| = |(V 0 ∩ v⊥) ∪ {0}| = 22d−2. (iii) For v ∈ V 1 we have |V 1 \ v⊥| = 22d−2 ∓V 2d−1. (iv) For v ∈ V 0 we have |V 1 \ v⊥| = 22d−2. (v) For v ∈ V 0 we have |V 1 ∩ v⊥| = 22d−2 ∓V 2d−1.

Proof.

Simple: One-two-line proofs for each statement.

Groups and Topological Groups 2016 Dominik Barth

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SLIDE 56

Proof of the Theorem

Lemma

|B||C| is not divisible by 2d−1.

Groups and Topological Groups 2016 Dominik Barth

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SLIDE 57

Proof of the Theorem

Lemma

|B||C| is not divisible by 2d−1.

Proof.

Groups and Topological Groups 2016 Dominik Barth

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SLIDE 58

Proof of the Theorem

Lemma

|B||C| is not divisible by 2d−1.

Proof.

|C| = |V 0 ∩ c⊥| =

Groups and Topological Groups 2016 Dominik Barth

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SLIDE 59

Proof of the Theorem

Lemma

|B||C| is not divisible by 2d−1.

Proof.

|C| = |V 0 ∩ c⊥| = 22d−2 − 1 is odd.

Groups and Topological Groups 2016 Dominik Barth

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SLIDE 60

Proof of the Theorem

Lemma

|B||C| is not divisible by 2d−1.

Proof.

|C| = |V 0 ∩ c⊥| = 22d−2 − 1 is odd. Let dim U = 2a, dim W = 2b. Then

Groups and Topological Groups 2016 Dominik Barth

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SLIDE 61

Proof of the Theorem

Lemma

|B||C| is not divisible by 2d−1.

Proof.

|C| = |V 0 ∩ c⊥| = 22d−2 − 1 is odd. Let dim U = 2a, dim W = 2b. Then |B| = |U1 + W 1| = |U1||W 1| = (22a−1 ∓U 2a−1)(22b−1 ∓W 2b−1)

Groups and Topological Groups 2016 Dominik Barth

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SLIDE 62

Proof of the Theorem

Lemma

|B||C| is not divisible by 2d−1.

Proof.

|C| = |V 0 ∩ c⊥| = 22d−2 − 1 is odd. Let dim U = 2a, dim W = 2b. Then |B| = |U1 + W 1| = |U1||W 1| = (22a−1 ∓U 2a−1)(22b−1 ∓W 2b−1) is not divisible by 2d−1 (using a + b = d).

Groups and Topological Groups 2016 Dominik Barth

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SLIDE 63

Proof of the Theorem

Lemma

|B||C| is not divisible by 2d−1.

Proof.

|C| = |V 0 ∩ c⊥| = 22d−2 − 1 is odd. Let dim U = 2a, dim W = 2b. Then |B| = |U1 + W 1| = |U1||W 1| = (22a−1 ∓U 2a−1)(22b−1 ∓W 2b−1) is not divisible by 2d−1 (using a + b = d). Together: |B||C| is not divisible by 2d−1.

Groups and Topological Groups 2016 Dominik Barth

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SLIDE 64

Proof of the Theorem

Lemma

2d−1 divides |B ∩ Cg| for all g ∈ O±(2d, 2).

Groups and Topological Groups 2016 Dominik Barth

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SLIDE 65

Proof of the Theorem

Lemma

2d−1 divides |B ∩ Cg| for all g ∈ O±(2d, 2).

Proof.

Note that B ∩ Cg = B ∩ (V 0 ∩ c⊥)g = B ∩ (cg)⊥

Groups and Topological Groups 2016 Dominik Barth

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SLIDE 66

Proof of the Theorem

Lemma

2d−1 divides |B ∩ Cg| for all g ∈ O±(2d, 2).

Proof.

Note that B ∩ Cg = B ∩ (V 0 ∩ c⊥)g = B ∩ (cg)⊥ and decompose cg = x + y with x ∈ U, y ∈ W. Then:

Groups and Topological Groups 2016 Dominik Barth

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SLIDE 67

Proof of the Theorem

Lemma

2d−1 divides |B ∩ Cg| for all g ∈ O±(2d, 2).

Proof.

Note that B ∩ Cg = B ∩ (V 0 ∩ c⊥)g = B ∩ (cg)⊥ and decompose cg = x + y with x ∈ U, y ∈ W. Then: |B ∩ Cg| = |B ∩ (cg)⊥| = |{u + w | u ∈ U1, w ∈ W 1, ϕ(u + w, cg) = 0}|

Groups and Topological Groups 2016 Dominik Barth

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SLIDE 68

Proof of the Theorem

Lemma

2d−1 divides |B ∩ Cg| for all g ∈ O±(2d, 2).

Proof.

Note that B ∩ Cg = B ∩ (V 0 ∩ c⊥)g = B ∩ (cg)⊥ and decompose cg = x + y with x ∈ U, y ∈ W. Then: |B ∩ Cg| = |B ∩ (cg)⊥| = |{u + w | u ∈ U1, w ∈ W 1, ϕ(u + w, cg) = 0}| = |{(u, w) ∈ U1 × W 1 | ϕ(u, cg) + ϕ(w, cg) = 0}|

Groups and Topological Groups 2016 Dominik Barth

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SLIDE 69

Proof of the Theorem

Lemma

2d−1 divides |B ∩ Cg| for all g ∈ O±(2d, 2).

Proof.

Note that B ∩ Cg = B ∩ (V 0 ∩ c⊥)g = B ∩ (cg)⊥ and decompose cg = x + y with x ∈ U, y ∈ W. Then: |B ∩ Cg| = |B ∩ (cg)⊥| = |{u + w | u ∈ U1, w ∈ W 1, ϕ(u + w, cg) = 0}| = |{(u, w) ∈ U1 × W 1 | ϕ(u, cg) + ϕ(w, cg) = 0}| = |{(u, w) ∈ U1 × W 1 | ϕ(u, x + y) + ϕ(w, x + y) = 0}|

Groups and Topological Groups 2016 Dominik Barth

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SLIDE 70

Proof of the Theorem

Lemma

2d−1 divides |B ∩ Cg| for all g ∈ O±(2d, 2).

Proof.

Note that B ∩ Cg = B ∩ (V 0 ∩ c⊥)g = B ∩ (cg)⊥ and decompose cg = x + y with x ∈ U, y ∈ W. Then: |B ∩ Cg| = |B ∩ (cg)⊥| = |{u + w | u ∈ U1, w ∈ W 1, ϕ(u + w, cg) = 0}| = |{(u, w) ∈ U1 × W 1 | ϕ(u, cg) + ϕ(w, cg) = 0}| = |{(u, w) ∈ U1 × W 1 | ϕ(u, x + y) + ϕ(w, x + y) = 0}| = |{(u, w) ∈ U1 × W 1 | ϕ(u, x) + ϕ(w, y) = 0}|

Groups and Topological Groups 2016 Dominik Barth

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SLIDE 71

Proof of the Theorem

Lemma

2d−1 divides |B ∩ Cg| for all g ∈ O±(2d, 2).

Proof.

Note that B ∩ Cg = B ∩ (V 0 ∩ c⊥)g = B ∩ (cg)⊥ and decompose cg = x + y with x ∈ U, y ∈ W. Then: |B ∩ Cg| = |B ∩ (cg)⊥| = |{u + w | u ∈ U1, w ∈ W 1, ϕ(u + w, cg) = 0}| = |{(u, w) ∈ U1 × W 1 | ϕ(u, cg) + ϕ(w, cg) = 0}| = |{(u, w) ∈ U1 × W 1 | ϕ(u, x + y) + ϕ(w, x + y) = 0}| = |{(u, w) ∈ U1 × W 1 | ϕ(u, x) + ϕ(w, y) = 0}| = |U1 ∩ x⊥| · |W 1 ∩ y⊥| + |U1 \ x⊥| · |W 1 \ y⊥|.

Groups and Topological Groups 2016 Dominik Barth

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SLIDE 72

Proof of the Theorem

Lemma

2d−1 divides |B ∩ Cg| for all g ∈ O±(2d, 2).

Proof.

Note that B ∩ Cg = B ∩ (V 0 ∩ c⊥)g = B ∩ (cg)⊥ and decompose cg = x + y with x ∈ U, y ∈ W. Then: |B ∩ Cg| = |B ∩ (cg)⊥| = |{u + w | u ∈ U1, w ∈ W 1, ϕ(u + w, cg) = 0}| = |{(u, w) ∈ U1 × W 1 | ϕ(u, cg) + ϕ(w, cg) = 0}| = |{(u, w) ∈ U1 × W 1 | ϕ(u, x + y) + ϕ(w, x + y) = 0}| = |{(u, w) ∈ U1 × W 1 | ϕ(u, x) + ϕ(w, y) = 0}| = |U1 ∩ x⊥| · |W 1 ∩ y⊥| + |U1 \ x⊥| · |W 1 \ y⊥|. Building blocks = ⇒ this term is divisible by 2d−1.

Groups and Topological Groups 2016 Dominik Barth

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SLIDE 73

Summary and Outlook

Contradicting subsets for O±(2d, 2), d ≥ 4, and therefore for Sp(2d, 2)

Groups and Topological Groups 2016 Dominik Barth

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SLIDE 74

Summary and Outlook

Contradicting subsets for O±(2d, 2), d ≥ 4, and therefore for Sp(2d, 2) Still open: An, Sn, M24.

Groups and Topological Groups 2016 Dominik Barth

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SLIDE 75

Summary and Outlook

Contradicting subsets for O±(2d, 2), d ≥ 4, and therefore for Sp(2d, 2) Still open: An, Sn, M24. Unfortunately, contradicting subsets (probably) won’t help here: Non-existence of contradicting subsets for Sn of degree n(n − 1) for all n ≤ 100

Groups and Topological Groups 2016 Dominik Barth

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SLIDE 76

For Further Reading

Thank you for your attention! P . Lorimer. Finite projective planes and sharply 2-transitive subsets of finite groups.

  • Proc. 2nd internat. Conf. Theory of Groups, Canberra 1973, Lect. Notes Math.

372, 432-436 (1974)., 1974.

  • M. O’Nan.

Sharply 2-transitive sets of permutations.

  • Proc. Rutgers group theory year, 1983/84, 63-67 (1984)., 1984.
  • T. Grundhöfer and P

. Müller. Sharply 2-transitive sets of permutations and groups of affine projectivities.

  • Beitr. Algebra Geom., 50(1):143–154, 2009.

P . Müller and G. P . Nagy. On the non-existence of sharply transitive sets of permutations in certain finite permutation groups.

  • Adv. Math. Commun., 5(2):303–308, 2011.
  • D. B.

The non-existence of sharply 2-transitive sets of permutations in Sp(2d, 2) of degree 22d−1 ± 2d−1 arXiv:1602.05164, 2016