Abstract Algebra Theory and Applications - Computer Science ...

11The Structure of GroupsThe ultimate goal of group theory is to classify all groups up to isomorphism;that is, given a particular group, we should be able to match it up with aknown group via an isomorphism. For example, we have already proved thatany finite cyclic group of order n is isomorphic to Z n ; hence, we “know” allfinite cyclic groups. It is probably not reasonable to expect that we will everknow all groups; however, we can often classify certain types of groups ordistinguish between groups in special cases.In this chapter we will characterize all finite abelian groups. We shall alsoinvestigate groups with sequences of subgroups. If a group has a sequenceof subgroups, sayG = H n ⊃ H n−1 ⊃ · · · ⊃ H 1 ⊃ H 0 = {e},where each subgroup H i is normal in H i+1 and each of the factor groupsH i+1 /H i is abelian, then G is a solvable group. In addition to allowing usto distinguish between certain classes of groups, solvable groups turn out tobe central to the study of solutions to polynomial equations.11.1 Finite Abelian GroupsIn our investigation of cyclic groups we found that every group of prime orderwas isomorphic to Z p , where p was a prime number. We also determinedthat Z mn∼ = Zm × Z n when gcd(m, n) = 1. In fact, much more is true.Every finite abelian group is isomorphic to a direct product of cyclic groupsof prime power order; that is, every finite abelian group is isomorphic to agroup of the type× · · · × Z pαn . nZ pα 11190