Symmetric 2-Structures, a Classification

We classify symmetric 2-structures ( P , G 1 , G 2 , K ) , i.e. chain structures which correspond to sharply 2-transitive permutation sets ( E , Σ) satisfying the condition: “ ( ∗ ) ∀ σ , τ ∈ Σ : σ ∘ τ - 1 ∘ σ ∈ Σ ”. To every chain K ∈ K one can associate a reflection K ~ in K . Then (*) is equivalent to “ ( ∗ ∗ ) ∀ K ∈ K : K ~ ( K ) = K ” and one can define an orthogonality “ ⊥ ” for chains K , L ∈ K by “ K ⊥ L ⇔ K ≠ L ∧ K ~ ( L ) = L ”. The classification is based on the cardinality of the set of chains which are orthogonal to a chain K and passing through a point p of K . For one of these classes (called point symmetric 2-structures ) we proof that in each point there is a reflection and that the set of point reflections forms a regular involutory permutation set.