Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Abstract Algebra-Isomorphism

  1. Dec 18, 2009 #1
    1. The problem statement, all variables and given/known data

    Let A,B be normal sub-groups of a group G.

    Prove that:
    G/AnB is isomorphic to G/A*G/B

    Have no idea how to start...Maybe the second isom. theorem can help us...


    2. Relevant equations
    3. The attempt at a solution
  2. jcsd
  3. Dec 18, 2009 #2
    Use the internal characterization of direct products of groups: if [tex]G[/tex] has two normal subgroups [tex]H, K[/tex] such that [tex]HK = G[/tex] and [tex]H \cap K = 1[/tex], then [tex]G \cong H \times K[/tex].

    Also, the third isomorphism theorem may help you (if [tex]K \subset H[/tex] are both normal subgroups of [tex]G[/tex], then [tex]G/H \cong (G/K)/(H/K)[/tex]).
  4. Dec 18, 2009 #3
    Sry but I rly can't figure out the Latex text (I see it in black, and it's realy not clear)...
    If I understand what you're saying, then we don't have the right conditions to use "internal characterization of direct products of groups"...
    A,B are normal sub-groups of G and AB=G but who said AnB={1}? The isomorphism you've put afterwards is relevant only when G=A*B and it isn't the case///

    Am I wrong?

  5. Dec 18, 2009 #4
    I've managed to prove it...TNX a lot anyway...
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook