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!

Finding Automorphism Groups

  1. May 12, 2009 #1
    1. The problem statement, all variables and given/known data
    Is there a good method for finding automorphism groups? I am currently working on finding them for D4 and D5.


    2. Relevant equations



    3. The attempt at a solution
    I've only really looked hard at D4 and the only one I've found is the identity. I know you have to send element of the same order to each other and in D4 there's the identity, two elements of order 4 and the remaining 5 are of order 2. I've been trying to look at ways to send the elements of order 2 to each other and there are a lot of ways, but none of the ones I've done end up being homomorphisms. My gut instinct is that for both of these there is more than one automorphism, but maybe I'm wrong.
     
  2. jcsd
  3. May 12, 2009 #2

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Look at inner automorphisms. g->sgs^(-1). There is certainly more than the identity automorphism.
     
  4. May 12, 2009 #3
    Ah, yes. Thanks. So it would appear that Aut(D4) is in fact isomorphic to D4 itself...
     
  5. May 13, 2009 #4

    matt grime

    User Avatar
    Science Advisor
    Homework Helper

    You have to do slightly more work than that.

    First you must show that the map D_4 to Aut(D_4) sending g to the inner automorphism is an injection or not, which it need not be (there are no inner automorphisms of an Abelian group). Then you need to work out if it is a surjection or not. If you were to do the same for S_6, then there are famously automorphisms that are not inner.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Finding Automorphism Groups
  1. Automorphism Groups (Replies: 1)

  2. Automorphism Group (Replies: 2)

Loading...