Fully characteristic subgroup

In mathematics, a subgroup of a group is fully characteristic (or fully invariant) if it is invariant under every endomorphism of the group. That is, any endomorphism of the group takes elements of the subgroup to elements of the subgroup.

Every group has itself (the improper subgroup) and the trivial subgroup as two of its fully characteristic subgroups. Every fully characteristic subgroup is a strictly characteristic subgroup, and a fortiori a characteristic subgroup.

The commutator subgroup of a group is always a fully characteristic subgroup. More generally, any verbal subgroup is always fully characteristic. Interestingly, for any reduced free group, and, in particular, for any free group, the converse also holds — every fully characteristic subgroup is verbal.

See also characteristic subgroup.