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5 Notes on Jordan-Hölder 5 Notes on Jordan-H¨older Definition 5.1. A subnormal series of a group G is a sequence of subgroups: G = H0 ¸ H1 ¸ H2 ¸ ¢ ¢ ¢ ¸ Hm = 1 so that each subgroup is normal in the previous one (HiEHi¡1). A refinement is another subnormal series obtained by adding extra terms. Definition 5.2. A composition series is a subnormal series G = G0 > G1 > ¢ ¢ ¢ > Gn = 1 so that Gi is a maximal proper normal subgroup of Gi¡1 for 1 · i · n. The quotients Gi¡1=Gi are simple groups called the composition factors of G. A composition factor is a special case of a subquotient of G which is defined to be a quotient of a subgroup of G (i.e., H=N where N E H · G). It is obvious that a finite group has at least one composition series. The Jordan-H¨olderTheorem says that the composition factors are uniquely de- termined up to permutation. To prove this we need to recall two of the basic theorems of group theory. Suppose that N EG and H · G. Then the Second Isomorphism Theorem says that NH=N »= H=N \H. The Correspondence Theorem says that there is a 1-1 correspondence between fD j N \H ED EHg and fE j N EE ENHg NH EH ND N \ H given by E = ND and D = E \ H and furthermore, NH H E D »= »= E D N N \ H The Zassenhaus Lemma compares two arbitrary subquotients of G, say A¤=A and B¤=B where G ¸ A¤ D A and G ¸ B¤ D B. Taking H = A¤ \ B¤ in the Second Isomorphism Theorem and D = (A \ B¤)(A¤ \ B) in the 1 Correspondence Theorem we get the following diagram. A(A¤ \ B¤)(A¤ \ B¤)B j j j A¤ \ B¤ j AD j DB j ADB A \ B¤ A¤ \ B Since AD = A(A\B¤)(A¤ \B) = A(A¤ \B) and DB = (A\B¤)(A¤ \B)B = (A \ B¤)B we get: Lemma 5.3 (Zassenhaus Lemma). A(A¤ \ B¤) (A¤ \ B¤)B »= A(A¤ \ B) (A \ B¤)B Theorem 5.4 (Schreier Refinement Theorem). Any two subnormal se- ries for G have equivalent refinements where equivalent means the sequences of subquotients are isomorphic after permutation. Proof. Suppose that G = G0 ¸ G1 ¸ ¢ ¢ ¢ ¸ Gm G = H0 ¸ H1 ¸ ¢ ¢ ¢ ¸ Hn are two subnormal series of G. Then we refine the first series by inserting between each of the n pairs Gi ¸ Gi+1 the following: Gi = Gi+1(Gi \ H0) ¸ Gi+1(Gi \ H1) ¸ ¢ ¢ ¢ ¸ Gi+1(Gi \ Hn) = Gi+1 Note that the subquotient Gi=Gi+1 is replaced by the m subquotients G (G \ H ) i+1 i j (1) Gi+1(Gi \ Hj+1) Similarly, the second series can be refined to replace each of the m subquo- tients Hj=Hj+1 by the n subquotients H (G \ H ) j+1 i j (2) Hj+1(Gi+1 \ Hj) By the Zassenhaus Lemma (1) is isomorphic to (2) so these are equivalent refinements. Theorem 5.5 (Jordan-H¨olderTheorem). Any two composition series for G are equivalent. Proof. Delete any repetitions in the refinements given by the Schreier Re- finement Theorem. 2 R-modules of finite length Suppose that M is a left (or right) R-module. Then everything works as before using the same diagrams and proofs with multiplication replaced with addition. For example the Zassenhaus Lemma says: A + (A¤ \ B¤) (A¤ \ B¤) + B »= A + (A¤ \ B) (A \ B¤) + B The Schreier Refinement Theorem holds where “subnormal series” should be interpreted as “finite descending sequence of submodules.” Similarly, the Jordan-H¨olderTheorem holds where a composition series for M is a finite descending sequence M = M0 ⊃ M1 ⊃ ¢ ¢ ¢ ⊃ Mn = 0 where each subquotient Mi=Mi+1 is a simple R-module. The number n is called the length of M. If M does not have a composition series then it has infinite length. Theorem 5.6. An R-module M has finite length if and only if it satisfies both chain conditions with respect to submodules. Proof. If both chain conditions are satisfied then it is obvious that the module has a composition series. [The ascending chain condition implies that every nonzero submodule of M has a maximal submodule. Taking Mi+1 to be a maximal submodule of Mi, we eventually get Mn = 0 by the descending chain condition.] Now suppose that M has finite length, say n, i.e., there is a composition series of length n. Then any ascending or descending sequence of submodules of M can have at most n + 1 distinct submodules. [If there were a longer sequence there would be more than n nonzero subquotients. By Schreier there would be a refinement of the composition series with more than n nonzero quotients contradicting the definition of a composition series.] HW3.ex 01: Show that an abelian group has a composition series if and only if it is finite. HW3.ex 02: A subgroup H of G is called subnormal if it is a member of a subnormal series of G. Prove that a group G has a composition series if and only if both chain conditions hold for subnormal subgroups. 3.
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