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Commutator subgroup and center

  1. Feb 3, 2008 #1
    1. The problem statement, all variables and given/known data
    Please confirm that the center of a group always contains the commutator subgroup. I am pretty sure its true.


    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Feb 3, 2008 #2

    StatusX

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    It's not. That would imply commutators commute, which there's no good reason to expect, and it should be easy to find a counterexample.
     
  4. Feb 4, 2008 #3
    But my book has a theorem that says:

    "If N is a normal subgroup of G, then G/N is abelian if and only if C is contained in N"
    where C is the commutator subgroup.

    Clearly the center is normal and its quotient group is abelian!
     
  5. Feb 4, 2008 #4

    morphism

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    G/Z(G) is not necessarily abelian, e.g. Z(A_5) is trivial (since A_5 is simple), whence A_5/Z(A_5) =~ A_5, a non-abelian group.
     
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