1. The problem statement, all variables and given/known data Is there a good method for finding automorphism groups? I am currently working on finding them for D4 and D5. 2. Relevant equations 3. 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.