Finding Automorphism Groups for D4 and D5

[Kalinka35]

Homework Statement

Is there a good method for finding automorphism groups? I am currently working on finding them for D4 and D5.

Homework Equations

The Attempt at a Solution

I've only really looked hard at D4 and the only one I've found is the identity. I know you have to send element of the same order to each other and in D4 there's the identity, two elements of order 4 and the remaining 5 are of order 2. I've been trying to look at ways to send the elements of order 2 to each other and there are a lot of ways, but none of the ones I've done end up being homomorphisms. My gut instinct is that for both of these there is more than one automorphism, but maybe I'm wrong.

 

[Dick]

Look at inner automorphisms. g->sgs^(-1). There is certainly more than the identity automorphism.

 

[Kalinka35]

Ah, yes. Thanks. So it would appear that Aut(D4) is in fact isomorphic to D4 itself...

 

[matt grime]

You have to do slightly more work than that.

First you must show that the map D_4 to Aut(D_4) sending g to the inner automorphism is an injection or not, which it need not be (there are no inner automorphisms of an Abelian group). Then you need to work out if it is a surjection or not. If you were to do the same for S_6, then there are famously automorphisms that are not inner.