## Automorphisms

An automorphism is an isomorphism from a group G onto itself.

Example: Ifthenis an automorphism of the group of complex numbers under addition. We test the requirements one by one.

1. With

2. Ifthenandsois one to one.

3.is onto since ifthenand

4.

The mappingsandare similarly automorphisms. All these automorphisms are length preserving.

A very important automorphism is the inner automorphism,whereis some element ofThis is called the automorphism ofinduced by

The inner automorphism ofinduced by(rotation by) is shown below.

The set of inner automorphisms is a group, as is the set of automorphisms.