1. Not finding help here? Sign up for a free 30min 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!

Center of a group

  1. Nov 5, 2006 #1
    "If N is a cyclic subgroup that is normal in G with index n and Aut(N) has no element of order n then N is central."

    Is this true?

    I think it is, but I don't know how to go about proving it... anyone have a hint that could get me started?
  2. jcsd
  3. Nov 5, 2006 #2


    User Avatar
    Science Advisor
    Homework Helper

    well lets just follow our nose. if N is not in the center, then its elements do notcommute with everything. so conjugating N by other elements of G, altough it elaves N \inv ariant, does not always give the identity automorphism of N.

    hence N has some automorhisms given by these other elements.

    now what would the orders of such automorphisms be? surely since N is cyclic, elements of n itelf do act trivially on N by conjugation, so the conjugation action defines map from G to Aut(N) with N in the kernel.

    Thus the image of G/N in Aut(N) has image of roder dividing indexN = n.

    But it is not clear to me it must be cyclic, nor even of order n.

    but suppose n were say prime? then what?

    look for a counter example as a semidirect product of two cyclic groups.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Center of a group
  1. Group with no center (Replies: 2)

  2. An Group (Replies: 1)

  3. Center of a Group (Replies: 3)

  4. Group center properties (Replies: 11)