Characteristic and Fully Invariant Subgroups

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.

 

Figure 1: Mappings hierarchy