I'm trying to get a better handle on characteristic subgroups, and many nice examples are given with some sort of "natural" definition. For example, it's clear that the center, torsion subgroup, and commutator subgroup of a given group are all characteristic, just because of the way they are defined. How can we formalize this "naturality"? The latter two have functors associated to them, but I'm not entirely certain if that's the reason for the subgroups being characteristic (and there isn't a similar functor for the center of a group).


EDIT: Let me explain why I'm asking this question. It's useful to know how various characteristic subgroups interact with direct products, quotients, and other group constructions. Henry's construction below is perhaps the right formalism, but it isn't clear to me what extra that buys us. On the other hand, knowing that a certain subgroup arises as the image of a functor means that we ought to be able to use categorical considerations to determine some properties of this subgroup. So here are a few related questions:


1. Is it possible to use definable subsets to see how characteristic subgroups act with respect to direct product, quotients, or other group constructions?
2. What properties does the Comm functor on Grp have? What about the Tors functor on Ab? Furthermore, which properties does a functor have to have in order to define a characteristic subgroup? (Other than the obvious property that F(G) is a subgroup of G for all G!)



How can we formalize the naturality of certain characteristic subgroups?
The commutator subgroup of a group is given by a functor on the category whose objects are groups and whose morphisms are all homomorphisms. We can say the similar statement for the torsion subgroup of an abelian group, and this is why the subgroups are fully invariant and not just characteristic. The center is not fully invariant (see e.g., the dihedral group of order two times four), and it is only given by a functor on the [core](http://ncatlab.org/nlab/show/core) of Grp, i.e., the category whose objects are groups and whose morphisms are isomorphisms.


If you have an endofunctor on the category of (abelian) groups, whose morphisms are all homomorphisms, such that objects are taken to subobjects, then those subobjects are fully invariant (hence characteristic) subgroups. If you have the endofunctor that is only defined on the core of Grp (or Ab), such that objects are taken to subobjects in Grp (not in the core), then the subobjects are characteristic subgroups. Proof: look at what the functor does to an endomorphism of the source.


