Automorphism group

Definition

Let $G$ be a group. An automorphism of $G$ is a group isomorphism from $G$ to $G$. The automorphism group $\operatorname{Aut}(G)$ is the set of all automorphisms of $G$, with composition of maps as the operation.

Remarks

The identity map is the identity element of $\operatorname{Aut}(G)$, and the inverse of an automorphism is its inverse map. The automorphism group of $(\mathbb{Z}, +)$ has two elements, the identity and $n \mapsto -n$. For an abelian group $G$, the map $g \mapsto g^{-1}$ is an automorphism.

Explanations

L1 human

An automorphism is a way to rename the elements of a group so that all products stay correct. The set of all such renamings is again a group, because you can compose and undo renamings. For the integers, the only renamings are "do nothing" and "change all signs".