Are the orders of Aut(G) and Inn(G) infinite for an infinite group G?

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In summary, automorphisms are bijective homomorphisms of a group. There is an inner automorphism induced by a in G. The orders of Aut(G) and Inn(G) are not always infinite.
  • #1
STEMucator
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So I've been reading a bit about automorphisms today and I was wondering about something. I'm particularly talking about groups in this case, say a group G.

So an automorphism is a bijective homomorphism ( endomorphism if you prefer ) of a group G.

We use Aut(G) to denote the set of all automorphisms of G.

There is also an inner automorphism induced by a in G. The map f := G → G such that f(x) = axa-1 for all x in G is called the inner automorphism of G induced by a.

We use Inn(G) to denote the set of all inner automorphisms of G.

My question now :

If |G| is infinite, are the orders of Aut(G) and Inn(G) also infinite? It makes sense to me that they would be.
 
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  • #2
Zondrina said:
If |G| is infinite, are the orders of Aut(G) and Inn(G) also infinite?

No. It is easy to check that [itex]\mathrm{Aut}(\mathbb{Z}) \cong \mathbb{Z}/(2)[/itex] and that [itex]\mathrm{Inn}(\mathbb{Z}) = 0[/itex].
 
  • #3
Zondrina said:
If |G| is infinite, are the orders of Aut(G) and Inn(G) also infinite? It makes sense to me that they would be.
No, not always. The integers, Z, is a group under addition. |Z| is infinite, but there are only two automorphisms on Z, given by f(x)=x (identity map) and g(x)=-x, respectively.
 
  • #4
Erland said:
No, not always. The integers, Z, is a group under addition. |Z| is infinite, but there are only two automorphisms on Z, given by f(x)=x (identity map) and g(x)=-x, respectively.

Ahh yes, that makes sense now because every other map doesn't generate all of Z.

So f(x) = 2x wouldn't be an automorphism for example. While it is a homomorphism, it is not injective, only surjective I believe.

Short proof to make sure : So f : Z → Z is already well defined for us. So we assume f(x) = f(y) → 2x = 2y → 2(x-y) = 0. Since 2|0, clearly x = y and f is injective. Now notice that the n elements of Z get mapped to fewer elements, so f is not surjective. Easy to show it's a homomorphism.

So f(x) = x and g(x) = -x are the only things that could be in Aut(Z) since they are the only maps which turn out to be automorphisms.

As for Inn(Z) = 0. That confused me a bit.

EDIT : Wouldn't it be Inn(Z) = { 0 } ?
 
  • #5
Zondrina said:
As for Inn(Z) = 0. That confused me a bit.

EDIT : Wouldn't it be Inn(Z) = { 0 } ?

They mean the same thing. Both of them are shorthand for the statement that [itex]\mathrm{Inn}(\mathbb{Z})[/itex] is the trivial group.
 
  • #6
Zondrina said:
So f(x) = 2x wouldn't be an automorphism for example. While it is a homomorphism, it is not injective, only surjective I believe.

You switched the two around. It is injective but not surjective, which is what you proved next.
 
  • #7
micromass said:
You switched the two around. It is injective but not surjective, which is what you proved next.

Oh whoops, mistype. Thanks for noticing.

Thanks for clarifying jgens I think I understand this now.
 

1. What is an automorphism?

An automorphism is a mathematical concept that refers to a transformation or mapping of an object onto itself. In simpler terms, an automorphism is a way of rearranging or relabeling an object while preserving its essential properties.

2. What are some examples of automorphisms?

Some common examples of automorphisms include rotations, reflections, and translations in geometry. In algebra, a common example is the identity function, where every element is mapped to itself. Other examples can vary depending on the specific object or structure being studied.

3. How are automorphisms useful in mathematics?

Automorphisms play a crucial role in many branches of mathematics, including group theory, topology, and algebraic geometry. They help to identify symmetries and patterns within mathematical objects and are often used to prove theorems and solve problems.

4. What is the difference between an automorphism and an isomorphism?

An automorphism is a transformation of an object onto itself, while an isomorphism is a bijective mapping between two different objects that preserves their structure. In other words, automorphisms are self-maps, while isomorphisms are maps between different objects.

5. How do automorphisms relate to symmetry?

Automorphisms and symmetry are closely related concepts. In fact, an automorphism can be thought of as a symmetry of an object, as it preserves its essential properties. Symmetry is often used to classify objects and identify patterns, making automorphisms a useful tool in understanding the structure of an object.

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