# Definition:Group Automorphism

## Definition

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.

## Examples

### Constant Product on Real Numbers

Let $\struct {\R, +}$ denote the real numbers under addition.

Let $\alpha \in \R$ be a real number.

Let $f: \R \to \R$ be the real function defined as:

$\forall x \in \R: \map f x = \alpha x$

Then $f$ is a (group) automorphism if and only if $\alpha \ne 0$.

## Also see

• Results about group automorphisms can be found here.

## 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.