 
Summary: Betweenness preserving permutations.
We let
be the identity map of the set of points.
Definition. We say a permutation of the set of points is betweenness preserving if
[s(a, b)] = s((a), (b)) whenever a and b are distinct points.
We let
B
be the set of betweenness preserving permutations of the set of points.
Theorem. B is a subgroup of the group of permutations of the set of points.
Proof. Exercise.
Remark. We will use repeatedly the fact that if B and X is a set of points then
1
[[X]] = (1
)[X] = [X] = X.
Proposition. Suppose A is a set of points and B. Then
[b(A)] = b([A]).
Proof. Suppose y [b(A)]. Then there is x in b(A) such that y = (x). Since x b(A) there is a in A such
that s(a, x) A. Let b = (a) and note that b [A]. Since B we have [s(a, x)] = s((a), (x)) = s(b, y).
Were it the case that y [A] we would have x = 1
