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!Automorphisms groups and spaces

  1. Dec 5, 2014 #1
    Is there any example of an automorphisms group of a space that coincides with the space, i.e. a space that is its own automorphisms group?
     
  2. jcsd
  3. Dec 5, 2014 #2
    Yes, there are many such examples. if I'm not mistaken ##\mathrm{Aut}(S_n)=S_n## for all ##n## except ##2,6##. But you can check it by hand that ##\mathrm{Aut}(S_3)=S_3##. The groups with this property are called complete groups.
     
    Last edited: Dec 5, 2014
  4. Dec 5, 2014 #3
    Thanks, I was under the impression that the isometry group of ##S^3## was ##SO(4)##.
     
  5. Dec 5, 2014 #4
    ##S^3## is the ##3##-sphere (the ##3##-dimensional sphere embedded in ##\mathbb{R}^4##), while ##S_3## is the group of permutations on ##3## letters. Totally different things.

    Does your problem concern Lie groups or any arbitrary groups?
     
  6. Dec 5, 2014 #5
    Oh, sorry, I should have made clear I referred to continuous spaces and groups.
     
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