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# Group isomorphisms and automorphisms watch

1. What is the difference between a group isomorphism and an automorphism. If you could give a few examples and explain why they are a group iso/automorphism then I would greatly appreciate it.
What I don't quite get is what it means when it is said that a group automorphism is a group isomorphism onto itself.
2. (Original post by B_9710)
What is the difference between a group isomorphism and an automorphism. If you could give a few examples and explain why they are a group iso/automorphism then I would greatly appreciate it.
What I don't quite get is what it means when it is said that a group automorphism is a group isomorphism onto itself.
Let G and H be groups. Then a group isomorphism is a 1-1 and onto map that satisfies a number of conditions. These conditions are often informally summarized by saying that is "structure preserving". What that means when you unwind is that matches up the elements of G with the elements of H in such a way that G and H behave in exactly the same way; as if the elements of H had been "relabeled" to look like the elements of G.

Now, we've made no requirement that G and H be different in the first place. can be a map between G and G. In this case, it's called an automorphism. What this means is that there is a permutation of G (i.e. a relabeling of the elements of G) that means that G-relabelled looks just like G.

Here's an example of an automorphism from Wolfram MathsWorld.
3. (Original post by Gregorius)
Let G and H be groups. Then a group isomorphism is a 1-1 and onto map that satisfies a number of conditions. These conditions are often informally summarized by saying that is "structure preserving". What that means when you unwind is that matches up the elements of G with the elements of H in such a way that G and H behave in exactly the same way; as if the elements of H had been "relabeled" to look like the elements of G.

Now, we've made no requirement that G and H be different in the first place. latex]\phi[/latex] can be a map between G and G. In this case, it's called an automorphism. What this means is that there is a permutation of G (i.e. a relabeling of the elements of G) that means that G-relabelled looks just like G.

Here's an example of an automorphism from Wolfram MathsWorld.
So permutations are automorphisms. That makes sense.
4. (Original post by B_9710)
So permutations are automorphisms. That makes sense.
Ah, be careful. Some permutations of the elements of a group are automorphisms; but they have to be permutations that respect the structure of the group.
5. (Original post by Gregorius)
Ah, be careful. Some permutations of the elements of a group are automorphisms; but they have to be permutations that respect the structure of the group.
What about if we have a permutation which is an isomorphism. Can we then say that the permutation is an automorphism?
6. (Original post by B_9710)
What about if we have a permutation which is an isomorphism. Can we then say that the permutation is an automorphism?
Yes. A permutation of a set is simply a 1-1 onto map between the elements of the set.

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