Of course, we know that any model of neutral geometry is either euclidean or hyperbolic, so must have a group associated with it. We have not actually done so, but we can show that the groups of these geometries are quite different. They are certainly not isomorphic. But we can show directly that a neutral greometry has an associated group, and that this can be defined as that generated by "reflections". We shall develop the theory to the point where we can see why the euclidean and hyperbolic groups have a common flavour. We will require some basic facts about neutral geometry: the length of segment AB is defined, and denoted by n(A,B) each line L divides the plane into two sides. Points P,Q not on L lie on opposite sides if and only if the segment PQ meets L. if P is a point and L is a line, then there is a unique line through P perpendicular to L if P and Q are distinct points, then there is a unique point P' on the line PQ such that Q is between P and P', and P, P' are equidistant from Q. we have the (SAS) and (SSS) conditions for congruence- the former is an axiom B lies on the segment AC if and only if n(A,B)+n(B,C) = n(A,C), B lies beyond C on the line AC if and only if n(A,B) = n(B,C) +n(A,C), B lies beyond A on the line AC if and only if n(C,B) = n(A,B) +n(A,C), B is not on the line AC if and only if n(A,C) < n(A,B) + n(C,B), if P and Q are distinct points, then there is a unique point R on PQ equidistant from P and Q the line through R perpendicular to PQ is called the perpendicular bisector of PQ. if P,Q,R are distinct points, then there is a unique line through Q making equal angles with the rays QP and QR (cutting the segment PR if R is not on PQ, through R if R is on the ray PQ) this is called the bisector of <PQR. two distinct circles meet at most twice, If L is a line, we can now define a map rL, called n-reflection in L, as follows if P is on L then rL(P) = P, if P is not on L, then rL(P) = Q, where Q lies on the perpendicular from P to L, P and Q are on opposite sides of L, and P, Q are equidistant from L. Directly, we can see that rL has order two, and that rL(P) = P if and only if P is on L. Thus rL has the right sort of properties to be considered a geometrical reflection. What makes these maps useful is that they preserve geometrical structures rL preserves length rL preserves angle rL maps lines to lines rL maps segments to segments In fact, the first is sufficient. By (5), we have (SSS). If P,Q,R map to P',Q',R', then, as lengths are preserved, <PQR = <P'Q'R'. Also, by (6), the points of a segment or line can be characterized in terms of length. Unfortunately, there is no easy proof of the invariance of length without looking at the other concepts to some extent. After this remark about the fundamental nature of length, it is natural to make the definition A map of a neutral geometry is an isometry if it preserves length. The set of isometries is clearly a group. It will come as no surprise that we can show that it is generated by the n-reflections.

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