Abstract Algebra Theory and Applications - Computer Science ...

196 CHAPTER 11 THE STRUCTURE OF GROUPSExample 6. The seriesis a refinement of the seriesZ ⊃ 3Z ⊃ 9Z ⊃ 45Z ⊃ 90Z ⊃ 180Z ⊃ {0}Z ⊃ 9Z ⊃ 45Z ⊃ 180Z ⊃ {0}.The correct way to study a subnormal or normal series of subgroups,{H i } of G, is actually to study the factor groups H i+1 /H i . We say that twosubnormal (normal) series {H i } and {K j } of a group G are isomorphic ifthere is a one-to-one correspondence between the collections of factor groups{H i+1 /H i } and {K j+1 /K j }.Example 7. The two normal seriesof the group Z 60 are isomorphic sinceZ 60 ⊃ 〈3〉 ⊃ 〈15〉 ⊃ {0}Z 60 ⊃ 〈4〉 ⊃ 〈20〉 ⊃ {0}Z 60 /〈3〉 ∼ = 〈20〉/{0} ∼ = Z 3〈3〉/〈15〉 ∼ = 〈4〉/〈20〉 ∼ = Z 5〈15〉/{0} ∼ = Z 60 /〈4〉 ∼ = Z 4 .A subnormal series {H i } of a group G is a composition series if all thefactor groups are simple; that is, if none of the factor groups of the seriescontains a normal subgroup. A normal series {H i } of G is a principalseries if all the factor groups are simple.Example 8. The group Z 60 has a composition serieswith factor groupsZ 60 ⊃ 〈3〉 ⊃ 〈15〉 ⊃ 〈30〉 ⊃ {0}Z 60 /〈3〉 ∼ = Z3〈3〉/〈15〉 ∼ = Z5〈15〉/〈30〉 ∼ = Z2〈30〉/{0} ∼ = Z 2 .