# 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.[1][2] 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 *H* of a group *G* is called a **characteristic subgroup**, *H* char *G*, if for every automorphism *φ* of *G*, φ[*H*] ≤ *H* holds, i.e. if every automorphism of the parent group maps the subgroup to within itself.

Given *H* char *G*, every automorphism of *G* induces an automorphism of the quotient group, *G/H*, which yields a map Aut(*G*) → Aut(*G*/*H*).

If *G* has a unique subgroup *H* of a given (finite) index, then *H* is characteristic in *G*.

## Related concepts

### Normal subgroup

A subgroup of *H* that is invariant under all inner automorphisms is called normal; also, an invariant subgroup.

- ∀φ ∈ Inn(
*G*)： φ[*H*] ≤*H*

Since Inn(*G*) ⊆ Aut(*G*) 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
*H*be a nontrivial group, and let*G*be the direct product,*H*×*H*. Then the subgroups, {1} ×*H*and*H*× {1}, are both normal, but neither is characteristic. In particular, neither of these subgroups is invariant under the automorphism, (*x*,*y*) → (*y*,*x*), that switches the two factors. - For a concrete example of this, let
*V*be the Klein four-group (which is isomorphic to the direct product, ℤ_{2}× ℤ_{2}). Since this group is abelian, every subgroup is normal; but every permutation of the 3 non-identity elements is an automorphism of*V*, so the 3 subgroups of order 2 are not characteristic. Here V = {*e*,*a*,*b*,*ab*} . Consider H = {*e*,*a*} and consider the automorphism, T(*e*) =*e*, T(*a*) =*b*, T(*b*) =*a*, T(*ab*) =*ab*; then T(*H*) is not contained in*H*. - 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, {1, −1}, is characteristic, since it is the only subgroup of order 2.
- If
*n*is even, the dihedral group of order 2*n*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 *strictly characteristic subgroup*, or a *distinguished subgroup*, 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), *H*, of a group, *G* is a group remaining invariant under every endomorphism of *G*; that is,

- ∀φ ∈ End(
*G*)： φ[*H*] ≤*H*.

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.[3][4]

Every endomorphism of *G* induces an endomorphism of *G/H*, which yields a map End(*G*) → End(*G*/*H*).

### 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 *H* is a (fully) characteristic subgroup of *K*, and *K* is a (fully) characteristic subgroup of *G*, then *H* is a (fully) characteristic subgroup of *G*.

*H*char*K*char*G*⇒*H*char*G*.

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

*H*char*K*⊲*G*⇒*H*⊲*G*

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 *H* char *G* and *K* is a subgroup of *G* containing *H*, then in general *H* is not necessarily characteristic in *K*.

*H*char*G*,*H*<*K*<*G*⇏*H*char*K*

## 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, Sym(3) × ℤ/2ℤ, has a homomorphism taking (*π*, *y*) to ((1, 2)^{y}, 0), which takes the center, 1 × ℤ/2ℤ, into a subgroup of Sym(3) × 1, 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 *G* = S_{3} × ℤ_{2} (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 *G* is its second factor ℤ_{2}. Note that the first factor, S_{3}, contains subgroups isomorphic to ℤ_{2}, for instance {e, (12)} ; let *f*: ℤ_{2} → S_{3} be the morphism mapping ℤ_{2} onto the indicated subgroup. Then the composition of the projection of *G* onto its second factor ℤ_{2}, followed by *f*, followed by the inclusion of S_{3} into *G* as its first factor, provides an endomorphism of *G* under which the image of the center, ℤ_{2}, is not contained in the center, so here the center is not a fully characteristic subgroup of *G*.

### 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

## References

- Dummit, David S.; Foote, Richard M. (2004).
*Abstract Algebra*(3rd ed.). John Wiley & Sons. ISBN 0-471-43334-9. - Lang, Serge (2002).
*Algebra*. Graduate Texts in Mathematics. Springer. ISBN 0-387-95385-X. -
Scott, W.R. (1987).
*Group Theory*. Dover. pp. 45–46. ISBN 0-486-65377-3. -
Magnus, Wilhelm; Karrass, Abraham; Solitar, Donald (2004).
*Combinatorial Group Theory*. Dover. pp. 74–85. ISBN 0-486-43830-9.