8 CHAPTER 1 Consider the case of a cube. The full group of the symmetries includes the "antipodal map" r, which reflects each vertex through the center of the cube. (Thus (A)T = F and (C)r = H in Figure 1.2, for instance.) The reader should check that T does not correspond to any rotation. Note that there is no antipodal map for the regular tetrahedron, although it is true for thatfiguretoo that there are symmetries that are not rotations. In fact, in this case, the full group of symmetries is the full symmetric group on the vertex set, of order 4! = 24. H (t-f\ Figure 1.2 Let us compute the order of the rotation group R of a cube. After a rotation, face ABCD can coincide with any of the six faces of the original cube, and in each location, it can have any of the four rotational orientations. It follows that \R\ = 6 • 4 = 24. The full group of symmetries 5, on the other hand, has order 48. (We leave this as an exercise.) What are the 24 symmetries that are not rotations? Among these are the reflections in the nine planes of symmetry of the cube. These planes of symmetry are of two types: six that contain four vertices (for instance, the planes determined by B, D, G, E or by A, E, S, F) and three that are parallel to faces of the cube. A tenth nonrotational symmetry is the antipodal map T. The remaining 14 nonrotational symmetries are rather hard to visualize and we shall not discuss them further now. The product (composition) of each of the nine reflections with r yields a rotation of order 2. (The order of an element g of a group, denoted o(g), is the least positive integer n, if it exists, such that gn is the identity. If there is no such n, we say that g has infinite order and write o(g) = oo. Elements of order 2 are usually called involutions,) A good exercise is to count how many elements of each order there are in the rotation group of a cube. We shall briefly mention three more examples before proceeding with our study of groups in general. Thefirstexample is the "general linear" group GL(V), where V is a vector space. This is the group of all nonsingular (invertible) linear transfor- mations of V. It should be obvious that GL(V) c Sym(V) is, in fact, a group. Next we consider the "affine group" of the line. This is the set of all mappings on the real numbers R that are of the form x H ax + b, where a, b € R and a ^ O The reader should check that this really is a group. Our final example is the group associated with the Rubik cube puzzle. (We assume that the reader has some familiarity with this object.) Of the 54 colored squares on the surface of the cube, six may be viewed as never moving from their

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