Definition:Automorphism Group/Group

Theorem
The set of automorphisms of a group $$G$$ is itself a group, under composition of mappings, and is denoted $$\operatorname{Aut} \left({G}\right)$$.

Proof
An automorphism is an isomorphism $$\phi: G \to G$$ from an algebraic structure $$G$$ to itself.

Taking the group axioms in turn:

G0: Closure
The composite of isomorphisms is itself an isomorphism, as demonstrated here.

G1: Associativity
The operation of composition of epimorphisms preserves associativity, as shown here.

An isomorphism is by definition an epimorphism.

The Composite of Bijections is a bijection.

G2: Identity
The Identity Mapping is an Automorphism.

G3: Inverses
If $$\phi \in \operatorname{Aut} \left({G}\right)$$, then $$\phi$$ is bijective and an isomorphism.

Hence from Inverse Isomorphism, $$\phi^{-1}$$ is also bijective and an isomorphism.

So $$\phi^{-1} \in \operatorname{Aut} \left({G}\right)$$.