Results 1 to 2 of 2

Math Help - composition series

  1. #1
    Member
    Joined
    Sep 2008
    Posts
    166

    composition series

    If G is a finite group and H is a normal subgroup of G. Prove that there is a composition series of G, one of whose terms is H.

    Recall that every finite group has a composition series.
    Definition: A composition series for G is a series of subgroups of G: {1} is a normal subgroup of N_0 is a normal subgroup of N_1 is a normal subgroup of ... is a normal subgroup of N_k = G such that N_i is a normal subgroup of N_{i+1} and N_{i+1}/N_i is simple.

    Proof:
    G is a finite group, so G has a composition series. And H is a normal subgroup of G, by the definition of a composition series, H must be a term of the series.

    Does this complete the proof? If not, please help. Thank you.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    9
    Quote Originally Posted by dori1123 View Post
    If G is a finite group and H is a normal subgroup of G. Prove that there is a composition series of G, one of whose terms is H.
    Begin by noticing H\triangleleft G. If H is a maximal normal subgroup then we can find a composition series H_0 \triangleleft H_1  ... \triangleleft H_n = H. Which gives H_0 \triangleleft H_1  ... \triangleleft H_n = H \triangleleft G.

    If H is not a maximal normal subgroup then there is H_{(1)} so that H \triangleleft H_{(1)} \triangleleft G so that H_{(1)}\triangleleft G is a maximal normal subgroup. Then repeat the argument by finding intermediate normal subgroups between H and H_{(1)}. Thus, we can form H_0 \triangleleft H_1 \triangleleft ... \triangleleft H_n = H \triangleleft ... \triangleleft G.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Function Composition.
    Posted in the Algebra Forum
    Replies: 2
    Last Post: December 8th 2011, 12:04 PM
  2. Composition
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: March 27th 2010, 05:12 PM
  3. Taylor series, Function composition
    Posted in the Calculus Forum
    Replies: 0
    Last Post: March 10th 2010, 02:38 PM
  4. Replies: 1
    Last Post: November 4th 2009, 09:52 AM
  5. composition
    Posted in the Pre-Calculus Forum
    Replies: 4
    Last Post: August 6th 2007, 05:38 PM

Search Tags


/mathhelpforum @mathhelpforum