inner group automorphism vs. group automorphism
Is there a fundamental difference of inner automorphisms and other automorphims of a group $G$ (I'm not asking for definitions)? I don't know how to formulate my question better... Maybe it's not clear what I'm asking for.
$\endgroup$ 11 Answer
$\begingroup$Inner automorphisms are automorphisms of a particular form: an automorphism $i: G\rightarrow G$ is an inner automorphism iff for some $g\in G$, $i(x)=gxg^{-1}$ for all $x\in G$.
Not every automorphism is inner. Note that if $G$ is abelian, then $gxg^{-1}=x$ for all $x, g\in G$, so the only inner automorphism of $G$ is the identity. So any nontrivial automorphism of an abelian group is not inner. In particular, the automorphism $$x\mapsto-x$$ of the group $(\mathbb{Z}, +)$ is not inner.
Here's a neat observation. Each inner automorphism comes from an element of the group (namely, the element $g$ in the definition). Put another way, every $g\in G$ yields an inner automorphism $\alpha_g$ given by $\alpha_g(x)=gxg^{-1}$.
Now note that $\alpha_g(\alpha_h(x))=ghxh^{-1}g^{-1}=(gh)x(gh)^{-1}=\alpha_{gh}(x)$. This tells us two things:
The inner automorphisms of $G$ form a group $Inn(G)$ under composition, just like the automorphisms in general (technically we need to check associativity and the existence of inverses and the identity, but that's trivial).
There is a surjective homomorphism from $G$ to $Inn(G)$, given by $g\mapsto \alpha_g$. In particular, $\vert Inn(G)\vert\le\vert G\vert$. By contrast, the whole automorphism group can be much larger than the original group.
