Examples of Automorphisms

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 $x \not= y$ but g^-1xg = g^-1yg, you find that g and its inverse cancel and that x=y, contradicting your assumption.) And it is a homomorphism:

Conj(x)Conj(y) = g^-1xgg^-1yg = g^-1xyg = Conj(xy)

Conjugations are also called inner automorphisms.

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

Inv(x)Inv(y) = x^-1y^-1

Inv(xy) = y^-1x^-1

but the group commutes, so x^-1y^-1=y^-1x^-1 and Inv(x)Inv(y) = Inv(xy)

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).