An automorphism is an isomorphism from a group G onto itself.
Example: If
then
is an automorphism of the group of complex numbers under addition. We test the requirements one by one.
1. With
2. If
then
and
so
is one to one.
3.is onto since if
then
and
4.
The mappings
and
are similarly automorphisms. All these automorphisms are length preserving.
A very important automorphism is the inner automorphism,
where
is some element of
This is called the automorphism of
induced by
The inner automorphism of
induced by
(rotation by
) is shown below.
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The set of inner automorphisms is a group, as is the set of automorphisms.