Fully-characteristic subgroup

A subgroup of a group $G$ that is invariant with respect to all endomorphisms of $G$. The set of fully-characteristic subgroups forms a sublattice in the lattice of all subgroups. The commutator subgroup and the members of the lower central series in an arbitrary group are fully-characteristic subgroups. In addition, any verbal subgroup of a group is fully characteristic. The converse statement is true for free groups: Any fully-characteristic subgroup is verbal.

References

[1]	W. Magnus, A. Karrass, B. Solitar, "Combinatorial group theory: presentations of groups in terms of generators and relations" , Interscience (1966)