Automorphism Group

For a group $G$, the automorphism group of $G$, denoted $\operatorname{Aut}(G)$, is the set of all group isomorphisms $\varphi:G\to G$, with group operation given by composition

The identity element is $\mathrm{id}_G$, and inverses are given by inverse maps.

Automorphisms are “symmetries” of $G$ that preserve the group structure. Many constructions (e.g. semidirect products ) are parametrized by homomorphisms into $\operatorname{Aut}(G)$.

Examples:

• $\operatorname{Aut}(\mathbb{Z})\cong C_2$, since an automorphism is determined by where it sends $1$, and it must send $1$ to $\pm 1$.
• If $G=C_n$ is cyclic of order $n$, then $\operatorname{Aut}(G)\cong (\mathbb{Z}/n\mathbb{Z})^\times$ (units mod $n$).
• For $G\times H$, there are automorphisms that swap factors when $G\cong H$, and more generally automorphisms can mix factors in nontrivial ways.