Automorphism - Imported Knowlpedia Content

Let $\mathcal C$ be a category and $A$ an object of $\mathcal C$ .

An automorphism of $A$ is an isomorphism

f:A\longrightarrow A.

Equivalent characterizations

Equivalently, it is an endomorphism $f:A\to A$ for which there exists $g:A\to A$ such that

g\circ f = \mathrm{id}_A,\qquad f\circ g = \mathrm{id}_A,

where $\mathrm{id}_A$ is the identity morphism.

The set of automorphisms of $A$ is denoted $\mathrm{Aut}_{\mathcal C}(A)$ .

Group structure

With composition as the operation, $\mathrm{Aut}_{\mathcal C}(A)$ is a group:

Examples

$\mathbf{Set}$ : Automorphisms of a set $X$ are exactly the bijections $X\to X$ . Thus $\mathrm{Aut}_{\mathbf{Set}}(X)$ is the permutation group of $X$ .

$\mathbf{Grp}$ : Automorphisms of a group $G$ are group isomorphisms $G\to G$ . Example: $\mathrm{Aut}_{\mathbf{Grp}}(\mathbb Z)\cong\{\pm 1\}$ , since any automorphism is determined by where it sends $1$ .

$R\mathbf{-Mod}$ : Automorphisms of an $R$ -module $M$ are the invertible $R$ -linear maps $M\to M$ . For $M=R^n$ , $\mathrm{Aut}_R(R^n)$ identifies with the group $\mathrm{GL}_n(R)$ of invertible $n\times n$ matrices over $R$ .