Automorphism

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

Definition

An automorphism of $A$ is an isomorphism

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

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:

• identity element: $\mathrm{id}_A$,
• inverse: $f^{-1}$ (the inverse isomorphism).

Examples

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

2. $\mathbf{Grp}$: 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 $1$.

3. $R\mathbf{-Mod}$: 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 $R$.