Abstract Algebra Theory and Applications - Computer Science ...

11.2 SOLVABLE GROUPS 197Since Z 60 is an abelian group, this series is automatically a principal series.Notice that a composition series need not be unique. The seriesZ 60 ⊃ 〈2〉 ⊃ 〈4〉 ⊃ 〈20〉 ⊃ {0}is also a composition series.Example 9. For n ≥ 5, the seriesS n ⊃ A n ⊃ {(1)}is a composition series for S n since S n /A n∼ = Z2 and A n is simple.Example 10. Not every group has a composition series or a principal series.Suppose that{0} = H 0 ⊂ H 1 ⊂ · · · ⊂ H n−1 ⊂ H n = Zis a subnormal series for the integers under addition. Then H 1 must be ofthe form nZ for some n ∈ N. In this case H 1 /H 0∼ = nZ is an infinite cyclicgroup with many nontrivial proper normal subgroups.Although composition series need not be unique as in the case of Z 60 , itturns out that any two composition series are related. The factor groups ofthe two composition series for Z 60 are Z 2 , Z 2 , Z 3 , and Z 5 ; that is, the twocomposition series are isomorphic. The Jordan-Hölder Theorem says thatthis is always the case.Theorem 11.6 (Jordan-Hölder) Any two composition series of G areisomorphic.Proof. We shall employ mathematical induction on the length of the compositionseries. If the length of a composition series is 1, then G must be asimple group. In this case any two composition series are isomorphic.Suppose now that the theorem is true for all groups having a compositionseries of length k, where 1 ≤ k < n. LetG = H n ⊃ H n−1 ⊃ · · · ⊃ H 1 ⊃ H 0 = {e}G = K m ⊃ K m−1 ⊃ · · · ⊃ K 1 ⊃ K 0 = {e}be two composition series for G. We can form two new subnormal series forG since H i ∩ K m−1 is normal in H i+1 ∩ K m−1 and K j ∩ H n−1 is normal inK j+1 ∩ H n−1 :G = H n ⊃ H n−1 ⊃ H n−1 ∩ K m−1 ⊃ · · · ⊃ H 0 ∩ K m−1 = {e}G = K m ⊃ K m−1 ⊃ K m−1 ∩ H n−1 ⊃ · · · ⊃ K 0 ∩ H n−1 = {e}.