The familiar operation of conjugation g-1xg is an automorphism. It
obviously maps group elements to other group elements; it is one-to-one (if
you assume otherwise, that
g-1xg = g-1yg, you find that
g and its inverse cancel and that x=y, contradicting your assumption.)
And it is a homomorphism:
Of course for commutative groups conjugates are not very interesting:
g-1xg = g-1gx = 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
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-1xg caused by that element we have a homomorphic mapping from G to Inn(G).