In 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 true. Examples of characteristic subgroups include the commutator subgroup and the center of a group.

Definition

is called a characteristic subgroup; then write .

.

Basic properties

induces an automorphism of the quotient group.

of a given index.

Related concepts

Normal subgroup

that is invariant under all inner automorphisms is called normal; sometimes called an invariant subgroup.

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:

• be the direct product, that switches the two factors.

  be the Klein four-group. Since this group is abelian.

  In the quaternion group, is characteristic, since it is the only subgroup of order 2.

  is even, the dihedral group subgroups of index, all of which are normal. One of these is the cyclic subgroup, which is characteristic. The other two subgroups are dihedral; these are permuted by an outer automorphism of the parent group, and are therefore not characteristic.

Strictly characteristic subgroupStrictly invariant 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, strictly characteristic is equivalent to characteristic. This is not the case for infinite groups.

Fully characteristic subgroupFully invariant subgroup

For an even stronger constraint, a fully characteristic subgroup (also, fully invariant subgroup, is a subgroup H ≤ G that is invariant under every endomorphism (and not just every automorphism):

1. .

Every group has itself (the improper subgroup) and the trivial subgroup as two of its fully characteristic subgroups. The commutator subgroup of a group is always a fully characteristic subgroup.

, 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.

1. .

Moreover, while normality is not transitive, it is true that every characteristic subgroup of a normal subgroup is normal.

Similarly, while being strictly characteristic (distinguished) is not transitive, it is true that every fully characteristic subgroup of a strictly characteristic subgroup is strictly characteristic.

However, unlike normality, if .

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, , into a subgroup of , which meets the center only in the identity.

The relationship amongst these subgroup properties can be expressed as:

1. Normal ⇐ Characteristic ⇐ Strictly characteristic ⇐ Fully characteristic ⇐ Verbal

Examples

Finite example

(the group of order 12 that is the direct product of the symmetric group of order 6 and a cyclic group.

Cyclic groups

Every subgroup of a cyclic group is characteristic.

Subgroup functors

The derived subgroup (or commutator subgroup) of a group is a verbal subgroup. The torsion subgroup of an abelian group is a fully invariant subgroup.

Topological groups

The identity component of a topological group is always a characteristic subgroup.

See also

• Characteristically simple group

Notes

References