symmetries and symmetry groups

 The most common way to introduce groups is to discuss the symmetries of plane figures, i.e. euclidean transformations which map the figure to itself. From the kleinian view, there is no reason why we should restrict attention to euclidean transformations. Suppose that G is a group of transformations of a set S, so that G gives a geometry on S. In this geometry, a figure F is simply a subset of S. We may consider elements of G which map F to itself. Definition Suppose that G is a group of transformations of a set S, and that F is a subset of S. An element g of G is a G-symmetry of F if g(F) = F. The set of G-symmetries of F is denoted by S(F,G). This set contains much geometrical information about F, as we shall see. Theorem SG1 If G is a group of transformations of a set S, and F is a subset of S, then S(F,G) is a subgroup of G. The subgroup S(F,G) is called the G-symmetry group of F. If H is any subgroup of S, it also defines a geometry on S, and hence we have an H-symmetry group S(F,H). Of course, the symmetry groups are related. Theorem K3 If G is a group of transformations of a set S, H a subgroup of G, and F is a subset of S, then S(F,H) is a subgroup of S(F,G). Proof Since S(F,H) is a group, by Theorem SG1, it is enough to show that it is a subset of S(F,G). Suppose that h is in S(F,H). Then h(F) = F. Since H is a subgroup of G, we also have hεG. Thus hεS(F,G), as required. At the moment, we can offer ony a rather contrived example to show that the subgroups may be different, i.e. that S(F,H) may be a proper subgroup. Example 1 Let F be the ray {(x,0) : x ≥ 0} in R2. Then it is easy to see that S(F,E(2)) contains only e, the identity, and r, reflection in the real axis. On the other hand, S(F,S(2)) also contains the scalings sk(x) = kx, k > 0. Thus, S(F,E(2)) is a proper subgroup of S(F,S(2)). Recall that two subsets F, F' of S are G-congruent if there is an element g of G with g(F) = F'. Since the definitions of G-congruence and G-symmetry are somewhat similar, it is not surprising that the concepts are related. Theorem SG2 If F and F' are G-congruent subsets of S, then S(F,G) and S(F',G) are conjugate subgroups of G. Indeed, if gεG is such that F' = g(F), then S(F',G) = gS(F,G)g-1. In particular, we observe that the G-symmetry groups of G-congruent subsets will be isomorphic groups. Unfortunately, the converse is false. We can have non-congruent figures with identical (and hence isomorphic) symmetry groups. Example 2 In the picture on the right, ABCD is a rectangle, and P,Q,P,S are the mid-points of AB,BC,CD,DA respectively. Then the quadrilaterals ABCD and PQRS each have E(2)-symmetry group K = {e,h,v,r}, where e is the identity, h and v are the reflections in the lines SQ and PR, and r is the half-turn about X. On the other hand the Theorem does allow us to conclude that figures with non-isomorphic G-symmetry groups must be incongruent in the G-geometry.