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Finding the order of the set of inner autos of D_4

  1. Mar 8, 2017 #1
    1. The problem statement, all variables and given/known data
    Find the order of ##Inn (D_4)##, where ##D_4## is the set of symmetries of the square.

    2. Relevant equations


    3. The attempt at a solution
    Is the only way to this by brute force calculation of all of the inner automorphisms, and to see which are distinct?
     
  2. jcsd
  3. Mar 8, 2017 #2

    fresh_42

    Staff: Mentor

    You've just proven ##Inn(D_4) \cong D_4/Z(D_4)##. So what is the center of ##D_4##?
     
  4. Mar 8, 2017 #3
    I think that the center is ##\{R_0, R_{180} \}##. So the order of ##D_4 / Z(D_4)## is 4, which makes the order of ##Inn( D_4)## 4?
     
  5. Mar 8, 2017 #4

    fresh_42

    Staff: Mentor

    Would have been my guess, too, but this is no proof. And even if we know the order were four, which one of the two groups of order four is it? do you know a representation of ##D_4## as a semi-direct product or which are the normal subgroups of ##D_4##?
     
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