# Definition:Automorphism (Abstract Algebra)

## Definition

An automorphism is an isomorphism from an algebraic structure to itself.

This applies to the term isomorphism as used both in the sense of bijective homomorphism as well as that of an order isomorphism.

Hence an automorphism is a permutation which is either a homomorphism or an order isomorphism, depending on context.

### Semigroup Automorphism

Let $\left({S, \circ}\right)$ be a semigroup.

Let $\phi: S \to S$ be a (semigroup) isomorphism from $S$ to itself.

Then $\phi$ is a semigroup automorphism.

### Group Automorphism

Let $\struct {G, \circ}$ be a group.

Let $\phi: G \to G$ be a (group) isomorphism from $G$ to itself.

Then $\phi$ is a group automorphism.

### Ring Automorphism

Let $\struct {R, +, \circ}$ be a ring.

Let $\phi: R \to R$ be a (ring) isomorphism.

Then $\phi$ is a ring automorphism.

### Field Automorphism

Let $\struct {F, +, \circ}$ be a field.

Let $\phi: F \to F$ be a (field) isomorphism from $F$ to itself.

Then $\phi$ is a field automorphism.

### $R$-Algebraic Structure Automorphism

Let $\left({S, \ast_1, \ast_2, \ldots, \ast_n, \circ}\right)_R$ be an $R$-algebraic structure.

Let $\phi: S \to S$ be an $R$-algebraic structure isomorphism from $S$ to itself.

Then $\phi$ is an $R$-algebraic structure automorphism.

This definition continues to apply when $S$ is a module, and also when it is a vector space.

### Ordered Structure Automorphism

Let $\left({S, \circ, \preceq}\right)$ be an ordered structures.

Let $\phi: S \to S$ be an ordered structure isomorphism from $S$ to itself.

Then $\phi$ is an ordered structure automorphism.

## Linguistic Note

The word automorphism derives from the Greek morphe (μορφή) meaning form or structure, with the prefix iso- meaning equal.

Thus automorphism means self structure.