The familiar operation of conjugation *g*^{-1}*xg* is an automorphism. It
obviously maps group elements to other group elements; it is one-to-one (if
you assume otherwise, that
but
*g*^{-1}*xg* = *g*^{-1}*yg*, you find that
*g* and its inverse cancel and that *x*=*y*, contradicting your assumption.)
And it is a homomorphism:

Conjugations are also called inner automorphisms.

Of course for commutative groups conjugates are not very interesting:
*g*^{-1}*xg* = *g*^{-1}*gx* = *x*. But there is another operation which is an
automorphism for these groups: mapping elements to their own inverses.
Inverses are unique, so this is 1-1, and

but the group commutes, so

One automorphism of a group can be followed by another, yielding a third
automorphism. Since automorphisms are one-to-one they have inverses, and
there is the identity automorphism of mapping every group element to itself,
so the set of automorphisms of a group is itself a set under the operation of
doing successive automorphism of the group. This group is called *Aut*, and
has a subgroup *Inn* of the conjugations, since the application of the
homomorphism equation above shows that a conjugation followed by a conjugation
is a conjugation. If we map each element *g* of *G* to the conjugation
*g*^{-1}*xg* caused by that element we have a homomorphic mapping from *G* to
*Inn*(*G*).