1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Show no non-abelian group G such that Z(G)=Z2 exists satisfying the mapping

  1. Nov 23, 2012 #1
    1. The problem statement, all variables and given/known data

    Show that there is no non-abelian group [itex]G[/itex] such that [itex]Z(G)=\mathbb{Z}_2[/itex], which satisfies the short exact [itex]\mathbb{Z}_2\rightarrow G\rightarrow\mathbb{Z}_2^3[/itex].


    3. The attempt at a solution

    I have knowledge of group theory up through proofs of the Sylow theorems. I know the center is contained in every normal subgroup of G. [itex]\mathbb{Z}_2^3[/itex] has a seven subgroups of order 2 so I've been trying to use the correspondence theorem to get some idea of what this implies for the structure of G, but no luck so far. I've found several paths to the fact that G has no element of order 8, but that still leaves a lot of possibilities for its subgroup of order 8. Anyways I've been banging my head against this one for a while now, can anyone help me out with it? Thanks.

    Note: I want to prove this without resorting to the classification of groups of order 16.
     
    Last edited: Nov 23, 2012
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted



Similar Discussions: Show no non-abelian group G such that Z(G)=Z2 exists satisfying the mapping
  1. Show G abelian (Replies: 2)

  2. Showing G is abelian (Replies: 1)

Loading...