# 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.

A subgroup *H* of a group *G* is called a **characteristic subgroup** if for every automorphism *φ* of *G*, one has φ(*H*) ≤ *H*; then write ** H char G**.

It would be equivalent to require the stronger condition φ(*H*) = *H* for every automorphism *φ* of *G*, because φ^{−1}(*H*) ≤ *H* implies the reverse inclusion *H* ≤ φ(*H*).

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

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

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

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:

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.

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,

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*).

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.

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

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:

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.

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