Outer automorphism group being a quotient group

Question

What is the significance of ${\rm Out}(G)$ being a quotient group of ${\rm Inn}(G)$ in ${\rm Aut}(G)$?

Thanks a lot in advance.

Answer 1

$\text{Out}(G)$ is supposed to measure the "outer" automorphisms of $G$. That is, the automorphisms that aren't inner.

To do this, we start with the group of all automorphisms $\text{Aut}(G)$ and kill the inner automorphisms ($\text{Inn}(G)$) by quotienting them out.

That is, we consider the quotient group $\text{Aut}(G) \big / \text{Inn}(G)$, whose elements are (morally) all the automorphisms that aren't inner, which is exactly what we wanted to study.

As for why someone would be interested in the outer automorphisms at all, that's a separate question. It turns out they're very interesting, basically because they're the "difficult to understand" automorphisms.

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I hope this helps ^_^

Answer 2

Here's another perspective on $Out(G)$ that might be clearer: Consider the map $\phi:G\to Aut(G)$ by $\phi(g)= \phi_g$ such that $\phi_g(h)= g^{-1}hg$ for all $h\in G$. It's a good exercise to prove that $\phi_g$ is indeed an automorphism of $G$ for all $g\in G$ and that $\phi$ is a group homomorphism when $Aut(G)$ is considered a group under composition. Let's ask some questions about the map $\phi$:

What's the kernel of $\phi$? It's the elements $g\in G$ such that $\phi_g(h)= g^{-1}hg= h$ or $hg=gh$ for all $h\in G$. Thus we've shown ker$(\phi)$ is the center of the group $Z(G)$.

What's the image of $\phi$? By the first isomorphism theorem we can take the quotient $G/\text{ker}(\phi)= G/Z(G) \cong \text{Im}(\phi)$. We now define $Inn(G) := \text{Im}(\phi)$.

What's the cokernel of $\phi$? Recall the cokernel of $\psi: G\to H$ is defined as the quotient $H/\text{Im}(\psi)$. In this case we have Coker$(\phi) = Aut(G)/\text{Im}(\phi) = Aut(G)/Inn(G)$. We define $Out(G):= $Coker$(\phi)$. So if $\phi$ is an isomorphism then Im$(\phi)=Aut(G)$ and thus $Out(G)$ is trivial. On the other hand if $G$ is abelian such that ker$(\phi)=Z(G)=G$ then $Out(G)\cong Aut(G)$.

Ok now with the definitions out of the way we want some idea of what these groups mean. Inner automorphisms are preserve the conjugacy classes of $G$. As LeeMosher saliently points out not all automorphisms which fix conjugacy classes are inner! In any case, it makes sense that if $G$ is abelian (each element forms a conjugacy class on its own) the inner automorphisms are trivial since they have to map each element to itself. Outer automorphisms are cosets of automorphisms up to inner automorphisms.

To illustrate let's consider $Out(\mathbb Z^+)$. Since $\mathbb Z^+$ is abelian we have $Aut(\mathbb Z^+)\cong Out(\mathbb Z^+)$. It's a good problem to check that $\psi(a)=-a$ is the only nontrivial automorphism (and thus outer automorphism) of $\mathbb Z^+$. Thus $Out(\mathbb Z^+)\cong C_2$.

If $G= Q_8$ is the quaternion group, or $\langle i,j,k \, \lvert \, i^2=j^2=k^2=ijk\rangle$ the conjugacy classes are defined by $[\{1\},\{-1\},\{\pm i\}, \{\pm j\},\{\pm k\}]$. The outer automorphisms are the cosets consisting of maps which swap around conjugacy classes in ways that still produce automorphisms. You can check that the cosets are thus defined by maps which permute {i, j, k} in different ways. Thus $Out(Q_8)\cong S_3$.

As another example let's take $S_n$. Here the conjugacy classes are "shapes" of permutations in cycle notation (for instance all permutations of the form $(\cdot\cdot)(\cdot\cdot)$ are a conjugacy class). So what are the cosets of isomorphisms of $S_n$ up to maps which fix permutation shape? It turns out that the class is trivial except for when n=6 for which there are two classes. So $Out(S_n)\cong C_1$ when $n\neq 6$ and $Out(S_6)\cong C_2$.

To conclude, you should think about outer automorphisms as cosets or classes of automorphisms which are invariant under composition with inner automorphisms or conjugation maps. If the group is abelian these cosets correspond to individual automorphisms of the group. The outer automorphism group usually tells us all the different ways to swap around conjugacy classes of a group in a way that still produces an automorphism -- as HallaSurvivor mentions this ends up being the information we really care about when looking at the automorphisms of a group.