DEFINITIONS AND EXAMPLES OF GROUPS 11 We should mention that the "elementwise" calculations in the preceding proof are not typical of most of algebra. The proof of Theorem 1.6, in fact, could almost serve as a model of what algebra is not, or at least should not be, in the opinion of the author. One way to describe the operation (multiplication) in an abstract group G is via a multiplication table. This is a square array, with rows and columns labeled by the elements of G and where the position in row x and column y is occupied by the element xy. Generally, it is neither useful nor practical to actually write down a multiplication table for G, but we can think of G as being defined by such a table. One of the advantages of thinking about groups abstractly, as in Definition 1.4, is that it allows us to see that certain groups, perhaps defined very differently, are essentially "the same." Suppose, for example, that we rename all the elements of some group G, and that we use these new names to relabel the rows and columns of the multiplication table of G and also to replace the entries in the table. The result will be the multiplication table of a group that is not, in any essential respect, different from G. We can make this notion of "essential sameness" more precise, as follows. (1.7) DEFINITION. Let G and H be two groups and suppose 9 : G - H is a bijection. We say that 9 is an isomorphism if 9(xy)=9(x)9(y) for all x, y € G. We say that G and H are isomorphic, and we write G = H if an isomorphism between them exists. If 9 is an isomorphism from G to H, then 9 induces a match-up of the elements of G with the elements of H that causes their multiplication tables to coincide. To the extent that we view groups as being defined by their multiplication tables, we see that isomorphic groups are essentially "the same." All "group theoretic" questions will have the same answers in G and H. For example, each of G and H will have equal numbers of elements of any given order, and G will be abelian iff H is abelian. (A group is said to be abelian if all of its elements commute, if xy = yx for all elements x, y.) As a concrete example, consider the group R of rotations of a cube and S = Sym(4). (We write Sym(n) as a shorthand for Sym({l, 2 , . . . , n}).) We have seen that |J?| = 24 and, of course, \S\ = 4! = 24. In fact, we will see that R = 5, and so these differently constructed objects are group theoretically identical. (Note that R permutes eight objects, the vertices of a cube, and S permutes {1, 2, 3,4}. As permutation groups, therefore, R and S are quite different.) In the cube of Figure 1.2, there are four "major diagonals," AF, BE,CH, and DG. Each element of R corresponds to a rotation of the cube and each such rotation induces apermutation of these four diagonals. If wefixan assignment ofthe numbers 1, 2, 3, and 4 to the four diagonals, then each element of R determines a particular element of S = Sym(4). To see that the corresponding mapping 9 : R -» S is an isomorphism, we need to establish that 9 is a bijection. It is not very hard to

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