Prove the set of all automorphisms of a group is a group.

[ArcanaNoir]

Homework Statement

An isomorphism of a group onto itself is called an automorphism. Prove that the set of all automorphisms of a group is itself a group with respect to composition.

Homework Equations

To prove that this is a group I must show that it is closed on composition, there is an identity, and each element has an inverse, but proving something is a group isn't where the trouble lies. The trouble lies in reading the problem/understanding the terms.

The Attempt at a Solution

First let's consider the thing called "automorphism". Is this a mapping? Say, the identity mapping? What exactly is the thing called "automorphism"?

Second, what is the set of all automorphisms of a group? How many ways can you really list the group? Isn't there only one? I'm pretty confused about these meanings. I don't actually need help showing this is a group, I need help knowing what set looks like.

 

[Jimmy Snyder]

Are you studying from a book? If the book has a question on automorphisms, then it should also have a definition. Have you looked in the index?

 

[ArcanaNoir]

The explanation given is "An isomorphism of a group onto itself is called an automorphism."

 

[Jimmy Snyder]

The explanation given is "An isomorphism of a group onto itself is called an automorphism."

What does it say that an isomorphism is?

 

[ArcanaNoir]

G is isomorphic to H means there is an operation preserving bijection from G to H.

 

[Jimmy Snyder]

G is isomorphic to H means there is an operation preserving bijection from G to H.

OK, one last question and we can get to work. What is the definition given for composition?

 

[ArcanaNoir]

f(g(x)) ? I don't know what you're asking here. Standard composition.

 

[Jimmy Snyder]

f(g(x)) ? I don't know what you're asking here. Standard composition.

That's what I was looking for. In order to prove that the set of automorphisms of a group G is itself a group under composition, you need to show 4 things. The first of these is closure. In other words, if f is an automorphism and g is an automorphism, then fg is an automorphism. In order to show that fg is an automorphism you have to show that it is an operation preserving bijection from G to G. There are four things to show:
1. fg maps G to G.
2. fg is one to one.
3. fg is onto
4. fg preserves the group operation on G.

Start with number 1, and so on through number 4. You have a lot of facts you can use.
1. f maps G to G.
2. f is one to one.
3. f is onto.
4. f preserves the group operation on G.
5. g maps G to G.
6. g is one to one.
7. g is onto.
8. g preserves the group operation on G.

Once you have done this, you are not finished. You will have only shown closure, but the rest is similar.

 

[ArcanaNoir]

Okay, that helps. Thanks :)