Automorphism Group - Knowlpedia

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

(\varphi\psi)(x) := \varphi(\psi(x)).

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

Remarks

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.