Abstract Algebra Theory and Applications - Computer Science ...

144 CHAPTER 8 ISOMORPHISMSthat is, we just multiply elements in the first coordinate as we do in G andelements in the second coordinate as we do in H. We have specified theparticular operations · and ◦ in each group here for the sake of clarity; weusually just write (g 1 , h 1 )(g 2 , h 2 ) = (g 1 g 2 , h 1 h 2 ).Proposition 8.7 Let G and H be groups. The set G × H is a group underthe operation (g 1 , h 1 )(g 2 , h 2 ) = (g 1 g 2 , h 1 h 2 ) where g 1 , g 2 ∈ G and h 1 , h 2 ∈ H.Proof. Clearly the binary operation defined above is closed. If e G and e Hare the identities of the groups G and H respectively, then (e G , e H ) is theidentity of G × H. The inverse of (g, h) ∈ G × H is (g −1 , h −1 ). The factthat the operation is associative follows directly from the associativity of Gand H.□Example 7. Let R be the group of real numbers under addition. TheCartesian product of R with itself, R × R = R 2 , is also a group, in which thegroup operation is just addition in each coordinate; that is, (a, b) + (c, d) =(a + c, b + d). The identity is (0, 0) and the inverse of (a, b) is (−a, −b). Example 8.ConsiderZ 2 × Z 2 = {(0, 0), (0, 1), (1, 0), (1, 1)}.Although Z 2 × Z 2 and Z 4 both contain four elements, it is easy to see thatthey are not isomorphic since for every element (a, b) in Z 2 × Z 2 , (a, b) +(a, b) = (0, 0), but Z 4 is cyclic.The group G × H is called the external direct product of G and H.Notice that there is nothing special about the fact that we have used onlytwo groups to build a new group. The direct productn∏G i = G 1 × G 2 × · · · × G ni=1of the groups G 1 , G 2 , . . . , G n is defined in exactly the same manner. IfG = G 1 = G 2 = · · · = G n , we often write G n instead of G 1 × G 2 × · · · × G n .Example 9. The group Z n 2 , considered as a set, is just the set of all binaryn-tuples. The group operation is the “exclusive or” of two binary n-tuples.For example,(01011101) + (01001011) = (00010110).This group is important in coding theory, in cryptography, and in manyareas of computer science.