Characteristic subgroup
In mathematics, particularly in the area of abstract algebra known as group theory, a characteristic subgroup is a subgroup that is mapped to itself by every automorphism of the parent group. Because every conjugation map is an inner automorphism, every characteristic subgroup is normal; though the converse is not guaranteed. Examples of characteristic subgroups include the commutator subgroup and the center of a group.
Definition
A subgroup of a group is called a characteristic subgroup if for every automorphism of, one has ; then write .It would be equivalent to require the stronger condition = for every automorphism of, because implies the reverse inclusion.
Basic properties
Given, every automorphism of induces an automorphism of the quotient group, which yields a homomorphism.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 ', is one which is invariant under surjective endomorphisms. 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 G, is a subgroup H ≤ G that is invariant under every endomorphism of :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.
Verbal subgroup
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: