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

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

Sharply Transitive Sets — Definitions Definitions Let Ω be a finite set and S ⊆ Sym Ω . Groups and Topological Groups 2016 Dominik Barth

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 α g = β . Groups and Topological Groups 2016 Dominik Barth

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 α g = β . S is sharply 2-transitive, if S is sharply transitive on { ( ω 1 , ω 2 ) ∈ Ω 2 | ω 1 � = ω 2 } . Groups and Topological Groups 2016 Dominik Barth

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 α g = β . S is sharply 2-transitive, if S is sharply transitive on { ( ω 1 , ω 2 ) ∈ Ω 2 | ω 1 � = ω 2 } . Observation (Witt) Sharply 2-transitive subsets of S n correspond to projective planes of order n . Groups and Topological Groups 2016 Dominik Barth

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 α g = β . S is sharply 2-transitive, if S is sharply transitive on { ( ω 1 , ω 2 ) ∈ Ω 2 | ω 1 � = ω 2 } . Observation (Witt) Sharply 2-transitive subsets of S n correspond to projective planes of order n . Problem (Hard) Show the non-existence of sharply 2-transitive sets in S n . Groups and Topological Groups 2016 Dominik Barth

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 α g = β . S is sharply 2-transitive, if S is sharply transitive on { ( ω 1 , ω 2 ) ∈ Ω 2 | ω 1 � = ω 2 } . Observation (Witt) Sharply 2-transitive subsets of S n correspond to projective planes of order n . Problem (Easier) Show the non-existence of sharply 2-transitive sets in subgroups of S n . Groups and Topological Groups 2016 Dominik Barth

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 α g = β . S is sharply 2-transitive, if S is sharply transitive on { ( ω 1 , ω 2 ) ∈ Ω 2 | ω 1 � = ω 2 } . Observation (Witt) Sharply 2-transitive subsets of S n correspond to projective planes of order n . Problem (Lorimer, 1970s) Show the non-existence of sharply 2-transitive sets in 2-transitive subgroups of S n . Groups and Topological Groups 2016 Dominik Barth

Sharply Transitive Sets — Results Problem (Lorimer, 1970s) Show the non-existence of sharply 2-transitive sets in 2-transitive subgroups of S n . Groups and Topological Groups 2016 Dominik Barth

Sharply Transitive Sets — Results Problem (Lorimer, 1970s) Show the non-existence of sharply 2-transitive sets in 2-transitive subgroups of S n . Theorem (Lorimer, O’Nan, Grundhöfer, Müller, Nagy) Suppose G ≤ S n contains a sharply 2-transitive subset. Then: Groups and Topological Groups 2016 Dominik Barth

Sharply Transitive Sets — Results Problem (Lorimer, 1970s) Show the non-existence of sharply 2-transitive sets in 2-transitive subgroups of S n . Theorem (Lorimer, O’Nan, Grundhöfer, Müller, Nagy) Suppose G ≤ S n contains a sharply 2-transitive subset. Then: G ≤ AGL e ( F p ) , n = p e , or Groups and Topological Groups 2016 Dominik Barth

Sharply Transitive Sets — Results Problem (Lorimer, 1970s) Show the non-existence of sharply 2-transitive sets in 2-transitive subgroups of S n . Theorem (Lorimer, O’Nan, Grundhöfer, Müller, Nagy) Suppose G ≤ S n contains a sharply 2-transitive subset. Then: G ≤ AGL e ( F p ) , n = p e , or G = A n , n ≡ 0 , 1 mod 4, or Groups and Topological Groups 2016 Dominik Barth

Sharply Transitive Sets — Results Problem (Lorimer, 1970s) Show the non-existence of sharply 2-transitive sets in 2-transitive subgroups of S n . Theorem (Lorimer, O’Nan, Grundhöfer, Müller, Nagy) Suppose G ≤ S n contains a sharply 2-transitive subset. Then: G ≤ AGL e ( F p ) , n = p e , or G = A n , n ≡ 0 , 1 mod 4, or G = S n , or Groups and Topological Groups 2016 Dominik Barth

Sharply Transitive Sets — Results Problem (Lorimer, 1970s) Show the non-existence of sharply 2-transitive sets in 2-transitive subgroups of S n . Theorem (Lorimer, O’Nan, Grundhöfer, Müller, Nagy) Suppose G ≤ S n contains a sharply 2-transitive subset. Then: G ≤ AGL e ( F p ) , n = p e , or G = A n , n ≡ 0 , 1 mod 4, or G = S n , or G = M 24 . Groups and Topological Groups 2016 Dominik Barth

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

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

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

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

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 ( 2 d , 2 ) of degree 2 2 d − 1 ± 2 d − 1 . (proved by Grundhöfer-Müller) Groups and Topological Groups 2016 Dominik Barth

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 ( 2 d , 2 ) of degree 2 2 d − 1 ± 2 d − 1 . (proved by Grundhöfer-Müller) In this talk Contradicting subsets for all Sp ( 2 d , 2 ) , d ≥ 4 Groups and Topological Groups 2016 Dominik Barth

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

Contradicting Subsets Lemma (Müller, Nagy) Let S ⊆ Sym Ω be sharply transitive and B , C ⊆ Ω arbitrary. Then � | B ∩ C g | . | B || C | = g ∈ S Groups and Topological Groups 2016 Dominik Barth

Contradicting Subsets Lemma (Müller, Nagy) Let S ⊆ Sym Ω be sharply transitive and B , C ⊆ Ω arbitrary. Then � | B ∩ C g | . | B || C | = g ∈ S Proof. Double counting of the set { ( b , c , g ) ∈ B × C × S | c g = b } . Groups and Topological Groups 2016 Dominik Barth

Contradicting Subsets Lemma (Müller, Nagy) Let S ⊆ Sym Ω be sharply transitive and B , C ⊆ Ω arbitrary. Then � | B ∩ C g | . | B || C | = g ∈ S Proof. Double counting of the set { ( b , c , g ) ∈ B × C × S | c g = b } . Definition Subsets B , C ⊆ Ω are contradicting subsets (modulo k ) for G ≤ Sym Ω , if: Groups and Topological Groups 2016 Dominik Barth

Contradicting Subsets Lemma (Müller, Nagy) Let S ⊆ Sym Ω be sharply transitive and B , C ⊆ Ω arbitrary. Then � | B ∩ C g | . | B || C | = g ∈ S Proof. Double counting of the set { ( b , c , g ) ∈ B × C × S | c g = b } . Definition Subsets B , C ⊆ Ω are contradicting subsets (modulo k ) for G ≤ Sym Ω , if: k divides | B ∩ C g | for all g ∈ G , but Groups and Topological Groups 2016 Dominik Barth

Contradicting Subsets Lemma (Müller, Nagy) Let S ⊆ Sym Ω be sharply transitive and B , C ⊆ Ω arbitrary. Then � | B ∩ C g | . | B || C | = g ∈ S Proof. Double counting of the set { ( b , c , g ) ∈ B × C × S | c g = b } . Definition Subsets B , C ⊆ Ω are contradicting subsets (modulo k ) for G ≤ Sym Ω , if: k divides | B ∩ C g | for all g ∈ G , but k does not divide | B || C | . Groups and Topological Groups 2016 Dominik Barth