Subgroup Series

• A subnormal (subinvariant) series of a group is a finite sequence of subgroups of such that and is a normal subgroup of and and .

• A normal (invariant) series of is a finite sequence of normal subgroups of such that , and .

• (Fraleigh e35.23) If

  is a subnormal (normal ) series then

• A subnormal (normal) series is a refinement of a subnormal (normal) series of a group if . That is, if each is one of the .

• Two subnormal (normal) series and of the same group are isomorphic if there is a one-to-one correspondence between the collections of factor groups and such that the corresponding factor groups are isomorphic.

• (Fraleigh 35.11) Schreier Theorem Two subnormal (normal) series of a group have isomorphic refinements.

  • Proof: Consider two subnormal series and with and elements respectively.

    , form the chain

    That is, we insert groups between each and . Denote such that we have a new chain of groups of not necessarily distinct groups and where :

    Construct a similar series for , setting for .

    The Zassenhaus Lemma gives us

    Which proves the theorem for subnormal series. For normal series, we note how each and are also normal in by (Fraleigh 34.4)

• A subnormal series of a group is a composition series if all the factor groups are simple.

  A normal series of is a principal series if all the factor groups are simple.

• (Fraleigh 35.15) Jordan-Holder Theorem Any two composition (principal) series on are isomorphic.

  • A composition series is a factorization of a group into simple factor groups. Such a factorization is unique.
  • is simple if and only if is a maximal normal subgroup. For a composition series, each must a be a maximal normal subgroup of .
  • For a principal series, each must be a maximal normal subgroup of that is also normal on .

• (Fraleigh 35.16) If has a composition (principal) series, and if is a proper normal subgroup of , then there exists a composition (principal) series containing .

• (Fraleigh e35.27) Let be a composition series for group . Let be a simple group. The distinct groups among also form a composition series for .

  • Proof: By the Isomorphism Theorems we have the following

    Note that is simple. Therefore is isomorphic to either or .

    In either case, is simple which proves the theorem.

• (Fraleigh e35.28) Let be a composition series for . Let and be the canonical homomorphism. The distinct groups among form a composition series for .

  • Proof: Observe that the map defined by
    Is a homomorphism such that
    Therefore by the First Isomorphism Theorem
    And the proof from (Fraleigh e35.27) shows that the above quotient is simple. Therefore, the distinct groups among form a composition series.

• (Fraleigh e35.24) An infinite Abelian group has no composition series.

  • Idea: Let be the infinite Abelian group. Clearly is not a composition series as has a proper normal subgroup. By (Fraleigh e35.23), however, it is impossible to create a composition series using finite groups since the product of their orders is finite while

• A group is solvable if it has a composition series such that all factor groups are abelian.

  • For a solvable group, every composition series must have abelian factor groups .
  • (Fraleigh e35.25) A finite direct product of solvable groups is solvable.
  • (Fraleigh e35.26) Let . If is solvable then is solvable.
    • Proof: Consider a composition series for . The set forms a composition series for as shown below. Note that because is Abelian and so is so is Abelian. The Second and Third IsomorphismTheorems can be applied
      Which proves the theorem since is Abelian.
  • (Fraleigh e35.29) A homomorphic image of a solvable group is solvable.

    • Proof: Use the same argument as (Fraleigh e35.27) and (Fraleigh e35.28) using the canonical map and solvable composition series .

      Note that is Abelian. Therefore, the homomorphic image is solvable.

• Let be a group and be its center. Also, let be the center of , where .

  The series

  is called the ascending central series of .

Links §

• A First Course in Abstract Algebra 7th Edition by Fraleigh