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

  • Thread starter NasuSama
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In summary, we are trying to determine Aut(G)/Inn(G) for G = D4. We know that Inn(G) consists of 4 elements and Aut(G) consists of 8 bijective functions. The order of the quotient is 2, meaning there are 2 functions. However, we have found 4 outer automorphisms, which is confusing. According to a source, there are 4 elements in Aut(G) that are also in Inn(G), and 4 elements in Aut(G) that are not in Inn(G). This leads us to finding two cosets in Aut(G)/Inn(G). We can determine these sets by considering that there are four automorphisms that map y to y and four
  • #1
NasuSama
326
3

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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  • #2
What is the order of the quotient?
 
  • #3
micromass said:
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.
 
  • #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)##.
 
  • #6
So this means that I need to find two cosets. This is the implication of what the writer is trying to show.
 
  • #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:
  • #8
I wonder if the one I have is true for [itex]\mathrm{Out}(D_4)[/itex]
 

What is the group G = D4?

The group G = D4, also known as the dihedral group of order 8, is a mathematical group consisting of the symmetries of a square. It has 8 elements and is denoted by D4 or D8.

What is Aut(G)?

Aut(G) is the automorphism group of G, which consists of all the isomorphisms from G to itself. In other words, Aut(G) is the set of all possible ways to "map" G onto itself while preserving the group structure.

What is Inn(G)?

Inn(G) is the inner automorphism group of G, which consists of all the automorphisms of G that are induced by conjugation. In other words, Inn(G) is the set of all possible ways to "map" G onto itself by applying an element of G to each element of G.

What is the order of Aut(G) and Inn(G)?

The order of Aut(G) is equal to the number of elements in G, which in this case is 8. The order of Inn(G) is equal to the number of conjugacy classes in G, which is also 8 for D4.

How do you determine Aut(G)/Inn(G) for G = D4?

To determine Aut(G)/Inn(G) for G = D4, we need to first find the number of elements in Aut(G) and Inn(G). Since both have an order of 8, Aut(G)/Inn(G) will have an order of 1. Therefore, Aut(G)/Inn(G) = {e}, where e is the identity element of D4.

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