Find the inner and outermorphisms of a particular dihedral group Given that |Inn($D_8$)| = 8 and |Out($D_8$)| = 2 where Out($D_8$) = Aut($D_8$)/Inn($D_8$) and $D_8$ = {e,r,$r^2$,..,$r^7$,s,sr,...,$sr^7$} we want to find Inn($D_8$) and Out($D_8$). We know that Out($D_8$) is a cyclic group (prime order) and we can let Out($D_8$) = {f Inn($D_8$):$f \in $Aut($D_8$)}. We kow the identity element must be in Out($D_8$) and also another element of order 2. I get stuck here and not sure how to proceed to find Inn($D_8$) and Out($D_8$).

**Hint:** There is always a surjective homomorphism $G \to \mathrm{Inn}(G)$ given by sending $g$ to conjugation by $g$. If you figure out the kernel, $K$, of this homomorphism then $\mathrm{Inn}(D_8) \simeq D_8/K$ and $D_8/K$ will be pretty easy to understand.

For "finding" $\mathrm{Out}(D_8)$, if you are given that it has order $2$ then you know that it's the group $\mathbb Z/2$. Is there something else you need to find?