Inner and Outer automorphisms

Definition The center $Z(G)$ of a group $G$ is the group of elements $z$ such that $\forall g \in G,\,\, zg = gz$. It is abelian and a normal subgroup of $G$.

Definition The set of automorphisms (isomorphisms from $G$ to $G$) $Aut(G)$ is a group under composition.

Notation: We will often write $x^\alpha$ for applying automorphisms $\alpha(x)$ and $x^g = g^{-1} x g$.

Definition The inner automorphisms $\theta_g(x) = g^{-1} x g$ form a group $Inn(G)$. $$(x^{g})^{g'} = \theta_{g'}(\theta_g(x)) = (gg')^{-1} x g g' = \theta_{gg'}(x) = x^{gg'}.$$

Theorem $G/Z(G) \simeq Aut(g)$.
The kernel of the group homomorphism $\theta : G \to Aut(G)$ are the elements $g$ such that $\theta_g(x)=x^g=g^{-1}xg=x$, this is exactly $Z(G)$.

Lemma $N$ is a normal subgroup of $G$ if $gNg^{-1} \subseteq N$.
proof: since group elements are invertible we actually have the equalty $gNg^{-1} = N$ multiply by $g$ to get the original definition of normal.

Theorem $Inn(G) \unlhd Aut(G)$.
Let $\theta_g$ be some inner automorphism then $\forall \alpha$ we have $\alpha (\theta_g (\alpha^{-1}(x))) = \alpha^{-1}(g^{-1})x\alpha^{-1}(g) = \theta_{\alpha(g)}$.

Definition The outer automorphism group is the quotient $Out(G) = Aut(G)/Inn(g)$.