We have already seen that conjugations are automorphisms, and that normal
subgroups are self-conjugate, i.e. preserved by conjugations on the group. A
characteristic subgroup is one which is preserved by all automorphisms of the
group, and may be seen as a refinement of normal subgroups. To be clear, any
automorphism of *G* maps elements of the characteristic subgroup to a distinct
and possibly not the same element of that characteristic subgroup. The only
element which must map to itself is the identity, preserved by all
homomorphisms.

In turn, *fully invariant* subgroups are mapped into themselves by all
endormophisms of the group. Note use of the word ``into'' here, as opposed to
``onto''. For example all groups have the trivial endormophism of mapping all
elements to the identity; this does not preserve subgroups the same way
conjugation and automorphisms preserve normal and characteristic subgroups.
But an endormorphism will never map elements of a fully invariant subgroup to
elements not in the subgroup. We will see an example of such a subgroup in
the section on commutators.