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

  • Thread starter NasuSama
  • Start date
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1. Homework Statement

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].
 
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What is the order of the quotient?
 
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What is the order of the quotient?
Then, it must be 2. This means there are 2 functions, but I found 4, which is quite strange.
 
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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)##.
 
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So this means that I need to find two cosets. This is the implication of what the writer is trying to show.
 
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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.
 
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I wonder if the one I have is true for [itex]\mathrm{Out}(D_4)[/itex]
 

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