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Determine Aut(G)/Inn(G), given G = D4

  1. May 21, 2013 #1
    1. The problem statement, all variables and given/known data

    Determine [itex]Aut(G)/Inn(G)[/itex], given [itex]G = D4[/itex].

    2. The attempt at a solution

    Let [itex]D_4 = \{e, x, y, y^2, y^3, xy, xy^2, xy^3\}[/itex]

    Found that [itex]Inn(G) = \{\phi_e, \phi_x, \phi_y, \phi_{xy}\}[/itex] and [itex]\mathrm{Aut}(G)[/itex] consists of 8 bijective functions. But I can't find [itex]Aut(G)/Inn(G)[/itex].
  2. jcsd
  3. May 21, 2013 #2
    What is the order of the quotient?
  4. May 21, 2013 #3
    Then, it must be 2. This means there are 2 functions, but I found 4, which is quite strange.
  5. May 21, 2013 #4
  6. May 21, 2013 #5
    They're saying that there are 4 elements in ##Aut(G)## that are also in ##Inn(G)##. And that there are 4 elements in ##Aut(G)## that are not in ##Inn(G)##.
  7. May 21, 2013 #6
    So this means that I need to find two cosets. This is the implication of what the writer is trying to show.
  8. May 21, 2013 #7
    Let's give a good try to determine the set of outer automorphism.

    Well, there are four automorphisms that map [itex]y[/itex] to [itex]y[/itex] while there are other four automorphisms that map [itex]y[/itex] to [itex]y^3[/itex]. So the first four belong to one set while the remaining four belong to another set. Thus, we obtain two sets in [itex]\mathrm{Aut}(G)/\mathrm{Inn}(G)[/itex].

    Is this even right? I am assuming that is the answer.
    Last edited: May 21, 2013
  9. May 21, 2013 #8
    I wonder if the one I have is true for [itex]\mathrm{Out}(D_4)[/itex]
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