Characteristic subgroup

Definition

A subgroup H of a group G is called 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.

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.

Normal subgroup

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

φ[H] ≤ H, ∀φ ∈ Inn(G)

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, ℤ₂ × ℤ₂). 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 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 subgroups

A strictly characteristic subgroup, or a distinguished subgroup, which is invariant under surjective endomorphisms. For finite groups, surjectivity 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 subgroups

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[H] ≤ H, ∀φ ∈ End(G).

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 subgroups

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

Finite example

Consider the group G = S₃ × ℤ₂ (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 ℤ₂. Note that the first factor, S₃, contains subgroups isomorphic to ℤ₂, for instance {e, (12)} ; let f: ℤ₂ → S₃ be the morphism mapping ℤ₂ onto the indicated subgroup. Then the composition of the projection of G onto its second factor ℤ₂, 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, ℤ₂, 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.