Group of inner automorphisms is isomorphic to a quotient

In summary, we are trying to show that the quotient group ##G / Z(G)## is isomorphic to the group of inner automorphisms of ##G##, ##Inn(G)##, by establishing a surjective homomorphism ##G \twoheadrightarrow Inn(G)\;##. This can be done by defining a general element of ##Inn(G)## as ##\varphi_g(x) = gxg^{-1}## and setting ##\mu (g) = \varphi_g##. The kernel of ##\mu## is then shown to be ##Z(G)##, leading to the use of the fundamental homomorphism theorem.
  • #1
Mr Davis 97
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Homework Statement


Let ##G## be any group. Recall that the center of ##G##, or ##Z(G)## is ##\{ x \in G ~ | ~ xg =
gx, ~ \forall g \in G\}##. Show that ##G / Z(G)## is isomorphic to ##Inn(G)##, the group of inner automorphisms of ##G## by ##g##.

Homework Equations

The Attempt at a Solution


I am not sure where to get started. I know that I am trying to find a particular isomorphism, but not sure how to find what that isomorphism must be, or whether that map will go from ##G / Z(G)## to ##Inn(G)## or the other way around.
 
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  • #2
Can you establish a surjective homomorphism ##G \twoheadrightarrow Inn(G)\;##?
 
  • #3
fresh_42 said:
Can you establish a surjective homomorphism ##G \twoheadrightarrow Inn(G)\;##?
Define a general element of ##Inn(G)## to be ##\varphi_g(x) = gxg^{-1}##

Would ##\mu : G \rightarrow Inn(G)## where ##\mu (g) = \varphi_g## be a surjection?
 
  • #4
Mr Davis 97 said:
Define a general element of ##Inn(G)## to be ##\varphi_g(x) = gxg^{-1}##

Would ##\mu : G \rightarrow Inn(G)## where ##\mu (g) = \varphi_g## be a surjection?
Yes, because every inner automorphism (=conjugation) looks like a ##\mu (g)##, so ##g## is the pre-imange. Now what is the kernel of ##\mu##?
 
  • #5
fresh_42 said:
Yes, because every inner automorphism (=conjugation) looks like a ##\mu (g)##, so ##g## is the pre-imange. Now what is the kernel of ##\mu##?
Scratch that

##Ker( \mu ) = Z(G)##
 
  • #6
No. ##\{e\} \subseteq \ker \mu## but not necessarily the entire kernel. The kernel is defined as the set of all elements that maps to ##e'## in the codomain. Now what is this ##e'## then and what does it mean, that ##\mu ## maps an element ##g## on it?
 
  • #7
fresh_42 said:
No. ##\{e\} \subseteq \ker \mu## but not necessarily the entire kernel. The kernel is defined as the set of all elements that maps to ##e'## in the codomain. Now what is this ##e'## then and what does it mean, that ##\mu ## maps an element ##g## on it?
Sorry, I made a mistake and was too slow to correct. I think that ##Ker (\mu ) = Z(G)##

Will the fundamental homomorphism theorem be used?
 
  • #8
Not sure what this theorem is, but sounds right. How do you now, that ##\ker \mu = Z(G)\,##? This is the essential part of the proof, so you should drop a line on it.
 
  • #9
fresh_42 said:
Not sure what this theorem is, but sounds right. How do you now, that ##\ker \mu = Z(G)\,##? This is the essential part of the proof, so you should drop a line on it.
##\ker \mu = \{x \in G ~ | ~ \mu (x) = \mu (e) \} = \{x \in G ~ | ~ \varphi_x = \varphi_e \} = \{x \in G ~ | ~ \varphi_x (g) = \varphi_e (g), ~ \forall g \in G \} = \{x \in G ~ | ~ x g x^{-1} = g, ~ \forall g \in G \} = \{x \in G ~ | ~ xg = gx, ~ \forall g \in G\} = Z(G)##
 
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Related to Group of inner automorphisms is isomorphic to a quotient

1. What is a group of inner automorphisms?

A group of inner automorphisms is a group that consists of automorphisms of a group that are defined by conjugation by elements of the group itself.

2. What does it mean for a group of inner automorphisms to be isomorphic to a quotient?

When a group of inner automorphisms is isomorphic to a quotient, it means that the group is structurally similar to a quotient group, meaning that the internal structure and operations are the same.

3. How do you prove that a group of inner automorphisms is isomorphic to a quotient?

To prove that a group of inner automorphisms is isomorphic to a quotient, you would typically use the isomorphism theorem, which states that if there is a homomorphism between two groups that is both injective and surjective, then the two groups are isomorphic.

4. Why is it important to study the isomorphism between a group of inner automorphisms and a quotient?

Studying the isomorphism between a group of inner automorphisms and a quotient can provide valuable insights into the structure and properties of both groups. It can also help us understand how different groups are related and how they can be used in various mathematical applications.

5. Can a group of inner automorphisms be isomorphic to more than one quotient?

Yes, it is possible for a group of inner automorphisms to be isomorphic to multiple quotient groups. This is because there can be multiple ways to define a homomorphism between two groups, and each homomorphism can result in a different isomorphism between the two groups.

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