Betweenness preserving permutations. be the identity map of the set of points. Summary: Betweenness preserving permutations. We let  be the identity map of the set of points. De nition. 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  2 B and X is a set of points then  1 [ [X ]] = ( 1 Æ  )[X ] = [X ] = X: Proposition. Suppose A is a set of points and  2 B. Then  [b(A)] = b( [A]): Proof. Suppose y 2  [b(A)]. Then there is x in b(A) such that y = (x). Since x 2 b(A) there is a in A such that s(a; x)  A. Let b = (a) and note that b 2  [A]. Since  2 B we have  [s(a; x)] = s((a); (x)) = s(b; y). Were it the case that y 2  [A] we would have x =  1 (y) 2  1 [ [A]] = A which is impossible since x 2 b(A) implies x 62 A. Thus y 2 b( [A]). Suppose y 2 b( [A]). Then y 62  [A] and there is b in  [A] such that s(b; y)   [A]. Let x =  1 (y) and Collections: Mathematics