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.
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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 , are both normal, but neither is characteristic. In particular, neither of these subgroups is invariant under the automorphism, , that switches the two factors.
* For a concrete example of this, let be the Klein four-group (which is isomorphic to the direct product, $\mathbb_2 \times \mathbb_2$). Since this group is abelian, every subgroup is normal; but every permutation of the 3 non-identity elements is an automorphism of , so the 3 subgroups of order 2 are not characteristic. Here . Consider and consider the automorphism, ; then is not contained in .
* 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, , is characteristic, since it is the only subgroup of order 2.
* If is even, the dihedral group of order 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 subgroup

A ', or a ', 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'' (also, ''fully invariant subgroup''; cf. invariant subgroup), , of a group , is a group remaining invariant under every endomorphism of ; that is,
:.

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

Moreover, while normality is not transitive, it is true that every characteristic subgroup of a normal subgroup is normal.
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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 and is a subgroup of containing , then in general is not necessarily characteristic in .
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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, $1 \times \mathbb / 2 \mathbb$, into a subgroup of , 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 (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 is isomorphic to its second factor $\mathbb_2$. Note that the first factor, , contains subgroups isomorphic to $\mathbb_2$, for instance ; let $f: \mathbb_2<\rarr \text_3$ be the morphism mapping $\mathbb_2$ onto the indicated subgroup. Then the composition of the projection of onto its second factor $\mathbb_2$, followed by , followed by the inclusion of into as its first factor, provides an endomorphism of under which the image of the center, $\mathbb_2$, 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.

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

References

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Subgroup properties