Abstract Algebra Theory and Applications - Computer Science ...

9.2 GROUP HOMOMORPHISMS 155In general, the subgroup nZ of Z is normal. The cosets of Z/nZ arenZ1 + nZ2 + nZ.(n − 1) + nZ.The sum of the cosets k + Z and l + Z is k + l + Z. Notice that we havewritten our cosets additively, because the group operation is integer addition.Example 5. Consider the dihedral group D n , generated by the two elementsr and s, satisfying the relationsr n = ids 2 = idsrs = r −1 .The element r actually generates the cyclic subgroup of rotations, R n , ofD n . Since srs −1 = srs = r −1 ∈ R n , the group of rotations is a normalsubgroup of D n ; therefore, D n /R n is a group. Since there are exactly twoelements in this group, it must be isomorphic to Z 2 .9.2 Group HomomorphismsOne of the basic ideas of algebra is the concept of a homomorphism, a naturalgeneralization of an isomorphism. If we relax the requirement that anisomorphism of groups be bijective, we have a homomorphism. A homomorphismbetween groups (G, ·) and (H, ◦) is a map φ : G → H suchthatφ(g 1 · g 2 ) = φ(g 1 ) ◦ φ(g 2 )for g 1 , g 2 ∈ G. The range of φ in H is called the homomorphic image of φ.Two groups are related in the strongest possible way if they are isomorphic;however, a weaker relationship may exist between two groups. Forexample, the symmetric group S n and the group Z 2 are related by the factthat S n can be divided into even and odd permutations that exhibit a groupstructure like that Z 2 , as shown in the following multiplication table.even oddeven even oddodd odd even