Fully invariant subgroup

Equivalent definitions in tabular format
A subgroup of a group is termed fully invariant or fully characteristic if it satisfies the following equivalent conditions:

Extreme examples

 * 1) The trivial subgroup is always fully invariant.
 * 2) Every group is fully invariant as a subgroup of itself.

Examples

 * 1) High occurrence example: In a cyclic group, every subgroup is fully invariant. That's because any subgroup can be described as the set of all $$d^{th}$$ powers, for some choice of $$d$$, and such a set is clearly invariant under endomorphisms. (In fact, it is a verbal subgroup).
 * 2) More generally, in any abelian group, the set of $$d^{th}$$ powers is a verbal subgroup, and hence fully invariant. The set of elements whose order divides $$d$$ is also fully invariant, though not necessarily verbal (for instance, in the group of all roots of unity, the subgroup of $$n^{th}$$ roots for fixed $$n$$ is fully invariant but not verbal).
 * 3) In a (possibly) non-abelian group, certain subgroup-defining functions always yield a fully invariant subgroup. For instance, the derived subgroup is fully invariant, and so are all terms of the lower central series as well as the derived series.

Non-examples

 * 1) In an elementary abelian group, and more generally, in a characteristically simple group, there is no proper nontrivial fully invariant subgroup (in fact, there's no proper nontrivial characteristic subgroup, either).
 * 2) There do exist characteristic subgroups that are not fully invariant; in fact, the center, and terms of the upper central series, may be characteristic but not fully invariant.

Formalisms
The property of being fully invariant has a second-order description. A subgroup $$H$$ of a group $$G$$ is termed fully characteristic if:

$$\forall \ g \in H, \forall \sigma \in G^G : \ (\ \forall \ a,b \in G, \sigma(ab) = \sigma(a)\sigma(b)) \implies \sigma(g) \in H$$

The condition in parentheses is a verification that the function $$\sigma$$ is an endomorphism of $$G$$.

Testing
Note that this GAP testing function uses an additional package called the SONATA package.

History
The concept was introduced by Levi in 1933 under the German name vollinvariant (translating to fully invariant). Both the terms fully invariant and fully characteristic are now in vogue.

Resolution of questions that are easy to formulate
Any typical question about the behavior of fully invariant subgroups in arbitrary groups that is easy to formulate will also be easy to resolve either with a proof or a counterexample, unless some other feature of the question significantly complicates it. This is so, despite the fact that there are a large number of easy-to-formulate questions about the endomorphism monoid that are still open. The reason is that even though not enough is known about the endomorphism monoids, there are other ways to obtain information about the structure of fully invariant subgroups.

At the one extreme, there are abelian groups, where the fully invariant subgroups are quite easy to get a handle on. At the other extreme, there are "all groups" where very little can be said about characteristic subgroups beyond what can be proved through elementary reasoning. The most interesting situation is in the middle, for instance, when we are looking at nilpotent groups and solvable groups. In these cases, there are some restrictions on the structure of fully invariant subgroups, but the exact nature of the restrictions is hard to work out.

Textbook references

 * , Page 28, Characteristic and fully invariant subgroups