A subgroup of a group is called a characteristic subgroup,, if for every automorphism of, holds, i.e. if every automorphism of the parent group maps the subgroup to within itself: Given, every automorphism of induces an automorphism of the quotient group,, which yields a map. If has a unique subgroup of a given index, then is characteristic in.
Related concepts
Normal subgroup
A subgroup of that is invariant under all inner automorphisms is called normal; also, an invariant subgroup. Since and a characteristic subgroup is invariant under all automorphisms, every characteristic subgroup is normal. However, not every normal subgroup is characteristic. Here are several examples:
Let be a nontrivial group, and let be the direct product,. Then the subgroups, and
Strictly characteristic subgroup
A ', or a ', which is invariant under surjectiveendomorphisms. For finite groups, surjectivity of an endomorphism implies injectivity, so a surjective endomorphism is an automorphism; thus being strictly characteristic is equivalent to characteristic. This is not the case anymore for infinite groups.
Fully characteristic subgroup
For an even stronger constraint, a fully characteristic subgroup,, of a group, is a group remaining invariant under every endomorphism of ; that is, Every group has itself and the trivial subgroup as two of its fully characteristic subgroups. The commutator subgroup of a group is always a fully characteristic subgroup. Every endomorphism of induces an endomorphism of, which yields a map.
An even stronger constraint is verbal subgroup, which is the image of a fully invariant subgroup of a free group under a homomorphism. More generally, any verbal subgroup is always fully characteristic. For any reduced free group, and, in particular, for any free group, the converse also holds: every fully characteristic subgroup is verbal.
Transitivity
The property of being characteristic or fully characteristic is transitive; if is a characteristic subgroup of, and is a characteristic subgroup of, then is a characteristic subgroup of. Moreover, while normality is not transitive, it is true that every characteristic subgroup of a normal subgroup is normal. Similarly, while being strictly characteristic is not transitive, it is true that every fully characteristic subgroup of a strictly characteristic subgroup is strictly characteristic. However, unlike normality, if and is a subgroup of containing, then in general is not necessarily characteristic in.
Containments
Every subgroup that is fully characteristic is certainly strictly characteristic and characteristic; but a characteristic or even strictly characteristic subgroup need not be fully characteristic. The center of a group is always a strictly characteristic subgroup, but it is not always fully characteristic. For example, the finite group of order 12,, has a homomorphism taking to, which takes the center,, into a subgroup of, which meets the center only in the identity. The relationship amongst these subgroup properties can be expressed as:
Consider the group . The center of is its second factor. Note that the first factor,, contains subgroups isomorphic to, for instance ; let be the morphism mapping onto the indicated subgroup. Then the composition of the projection of onto its second factor, followed by, followed by the inclusion of into as its first factor, provides an endomorphism of under which the image of the center,, is not contained in the center, so here the center is not a fully characteristic subgroup of.
Cyclic groups
Every subgroup of a cyclic group is characteristic.
