Characteristic subgroup

Template:Short description 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 Template:Math of a group Template:Math is called a characteristic subgroup if for every automorphism Template:Math of Template:Math, one has Template:Math; then write Template:Math.

It would be equivalent to require the stronger condition Template:Math = Template:Math for every automorphism Template:Math of Template:Math, because Template:Math implies the reverse inclusion Template:Math.

Basic properties

Given Template:Math, every automorphism of Template:Math induces an automorphism of the quotient group Template:Math, which yields a homomorphism Template:Math.

If Template:Math has a unique subgroup Template:Math of a given index, then Template:Math is characteristic in Template:Math.

Related concepts

Normal subgroup

Template:Main A subgroup of Template:Math that is invariant under all inner automorphisms is called normal; also, an invariant subgroup.

Template:Math

Since Template:Math 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 Template:Math be a nontrivial group, and let Template:Math be the direct product, Template:Math. Then the subgroups, Template:Math and Template:Math, are both normal, but neither is characteristic. In particular, neither of these subgroups is invariant under the automorphism, Template:Math, that switches the two factors.
• For a concrete example of this, let Template:Math be the Klein four-group (which is isomorphic to the direct product, ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$). Since this group is abelian, every subgroup is normal; but every permutation of the 3 non-identity elements is an automorphism of Template:Math, so the 3 subgroups of order 2 are not characteristic. Here Template:Math. Consider Template:Math and consider the automorphism, Template:Math; then Template:Math is not contained in Template:Math.
• In the quaternion group of order 8, each of the cyclic subgroups of order 4 is normal, but none of these are characteristic. However, the subgroup, Template:Math, is characteristic, since it is the only subgroup of order 2.
• If Template:Math > 2 is even, the dihedral group of order Template:Math has 3 subgroups of index 2, 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 subgroupTemplate:Anchor

A Template:Vanchor, or a Template:Vanchor, 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 subgroupTemplate:Anchor

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

Template:Math.

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.

Every endomorphism of Template:Math induces an endomorphism of Template:Math, which yields a map Template:Math.

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 Template:Math is a (fully) characteristic subgroup of Template:Math, and Template:Math is a (fully) characteristic subgroup of Template:Math, then Template:Math is a (fully) characteristic subgroup of Template:Math.

Template:Math.

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

Template:Math

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 Template:Math and Template:Math is a subgroup of Template:Math containing Template:Math, then in general Template:Math is not necessarily characteristic in Template:Math.

Template:Math

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, Template:Math, has a homomorphism taking Template:Math to Template:Math, which takes the center, $1\times \mathbb{Z}/2\mathbb{Z}$, into a subgroup of Template:Math, which meets the center only in the identity.

The relationship amongst these subgroup properties can be expressed as:

Subgroup ⇐ Normal subgroup ⇐ Characteristic subgroup ⇐ Strictly characteristic subgroup ⇐ Fully characteristic subgroup ⇐ Verbal subgroup

Examples

Finite example

Consider the group Template:Math (the group of order 12 that is the direct product of the symmetric group of order 6 and a cyclic group of order 2). The center of Template:Math is isomorphic to its second factor ${\mathbb{Z}}_2$. Note that the first factor, Template:Math, contains subgroups isomorphic to ${\mathbb{Z}}_2$, for instance Template:Math; let $f:{\mathbb{Z}}_2<\to {\text{S}}_3$ be the morphism mapping ${\mathbb{Z}}_2$ onto the indicated subgroup. Then the composition of the projection of Template:Math onto its second factor ${\mathbb{Z}}_2$, followed by Template:Math, followed by the inclusion of Template:Math into Template:Math as its first factor, provides an endomorphism of Template:Math under which the image of the center, ${\mathbb{Z}}_2$, is not contained in the center, so here the center is not a fully characteristic subgroup of Template:Math.

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