- #1

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## Homework Statement

Find all of the elements in ##\operatorname{Aut}(D_8)##.

## Homework Equations

## The Attempt at a Solution

I am trying to come with a repeatable procedure for finding automorphisms of groups with presentations. I'm taking ##D_8## as a good example.

Here is how I would like to start. Let ##\psi\in \operatorname{Aut}(D_8)##. Then by the universal mapping property for presentations, ##\psi## is completely determined by ##r'=\psi(r)## and ##s'=\psi(s)## which must satisfy ##(r')^4 = (s')^2 = r's'r's'=1##. Since ##\psi## is an automorphism, it must preserve order, so the possible automorphisms are given by ##\psi(r) \in \{r,r^{-1}\}## and ##\psi(s) \in \{r^2,s,sr,sr^2,sr^3\}##. But all of these may not be automorphisms, since we haven't checked if the relations are satisfied. Note we can exclude the case when ##\psi(r)=r^{\pm1}## and ##\psi(s) = r^2##, since this does not satisfy the relation ##r's'r's'=1##. So we are left with ##8## possible automorphisms. Note that ##\operatorname{im}(\psi)## contains the subgroup ##\langle r' \rangle## of order ##6## and at least one element ##s'## not in this group, so ##|\operatorname{im}(\psi)| \ge 5## and by Lagrange ##|\operatorname{im}(\psi)|## divides ##8##. Hence ##\operatorname{im}(\psi) = D_8##, so ##\psi## is surjective (and hence injective), so ##\psi## is a bijection. All that's left to show is that all of these ##8## possibilities are indeed homomorphisms...

This is where I get stuck. Do I need to show that the ##r's'r's'=1## for every choice of ##r'## and ##s'## to show that the relations hold and that we get homomorphisms for all of these? This seems like a lot of work.