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 H of a group G is called a if for every automorphism φ of G , one has φ(H) ≤ H 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) .

Basic properties

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 .

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: H be a nontrivial group, and let G be the direct product, H × H . Then the subgroups, {1} × H and {{math|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. V be the Klein four-group (which is isomorphic to the direct product, ). 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 . Consider {{math|H = {e, a{{)}}}} and consider the automorphism, {{math|T(e) = e, T(a) = b, T(b) = a, T(ab) {{=}} ab}}; then T(H) is not contained in H . n

2 is even, the dihedral group of order 2n 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 ∀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. 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) × , has a homomorphism taking (π, y) to ((1, 2)y, 0) , which takes the center,, 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:

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 G is isomorphic to its second factor. Note that the first factor, S3 , contains subgroups isomorphic to, for instance {e, (12)} G onto its second factor, followed by f , followed by the inclusion of S3 into G as its first factor, provides an endomorphism of G under which the image of the center,, 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.