# Characteristic subgroup

In mathematics, particularly in the area of abstract algebra known as group theory, a **characteristic subgroup** is a subgroup that is invariant under all automorphisms of the parent group.^{[1]}^{[2]} Because conjugation is an automorphism, every characteristic subgroup is normal, though not every normal subgroup is characteristic. Examples of characteristic subgroups include the commutator subgroup and the center of a group.

## Definitions

A **characteristic subgroup** of a group *G* is a subgroup *H* that is invariant under each automorphism of *G*. That is,

for every automorphism *φ* of *G* (where *φ*(*H*) denotes the image of *H* under *φ*).

The statement “*H* is a characteristic subgroup of *G*” is written

## Characteristic vs. normal

If *G* is a group, and *g* is a fixed element of *G*, then the conjugation map

is an automorphism of *G* (known as an inner automorphism). A subgroup of *G* that is invariant under all inner automorphisms is called normal. Since a characteristic subgroup is invariant under all automorphisms, every characteristic subgroup is normal.

Not every normal subgroup is characteristic. Here are several examples:

- Let
*H*be a 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 three non-identity elements is an automorphism of*V*, so the three subgroups of order 2 are not characteristic.Here Consider H={e,a} and consider the automorphism .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.

Note: If *H* is the unique subgroup of a group *G*, then *H* is characteristic in *G*.

- If
*n*is even, the dihedral group of order 2*n*has three subgroups of index two, 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. - "Normality" is not transitive, but Characteristic has a transitive property, namely if
*H*Char*K*and*K*normal in*G*then*H*normal in*G*.

## Comparison to other subgroup properties

### Distinguished subgroups

A related concept is that of a **distinguished subgroup** (also called **strictly characteristic subgroup**). In this case the subgroup *H* is invariant under the applications of surjective endomorphisms. For a finite group this is the same, because surjectivity implies injectivity, but not for an infinite group: a surjective endomorphism is not necessarily an automorphism.

### Fully invariant subgroups

For an even stronger constraint, a fully characteristic subgroup (also called a **fully invariant subgroup**) *H* of a group *G* is a group remaining invariant under every endomorphism of *G*; in other words, if *f* : *G* → *G* is any homomorphism, then *f*(*H*) is a subgroup of *H*.

### Verbal subgroups

An even stronger constraint is verbal subgroup, which is the image of a fully invariant subgroup of a free group under a homomorphism.

### Containments

Every subgroup that is fully characteristic is certainly distinguished and therefore characteristic; but a characteristic or even distinguished subgroup need not be fully characteristic.

The center of a group is always a distinguished subgroup, but it is not always fully characteristic. The finite group of order 12, Sym(3) × **Z**/2**Z** has a homomorphism taking (*π*, *y*) to ( (1,2)^{y}, *0*) which takes the center 1 × **Z**/2**Z** 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**⇐ distinguished subgroup ⇐ fully characteristic subgroup ⇐ verbal subgroup

## Examples

### Finite example

Consider the group *G* = S_{3} × Z_{2} (the group of order 12 which 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 {identity,(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.

## 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*.

Moreover, while it is not true that every normal subgroup of a normal subgroup is normal, it is true that every characteristic subgroup of a normal subgroup is normal. Similarly, while it is not true that every distinguished subgroup of a distinguished subgroup is distinguished, it is true that every fully characteristic subgroup of a distinguished subgroup is distinguished.

## Map on Aut and End

If , then every automorphism of *G* induces an automorphism of the quotient group *G/H*, which yields a map .

If *H* is fully characteristic in *G*, then analogously, every endomorphism of *G* induces an endomorphism of *G/H*, which yields a map
.