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 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).

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

Main article: 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, ${\displaystyle \mathbb {Z} _{2}\times \mathbb {Z} _{2}}$). 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 > 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) of a group G, is a subgroup H ≤ G that is invariant under every endomorphism of G (and not just every automorphism):

∀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) × ${\displaystyle \mathbb {Z} /2\mathbb {Z} }$, has a homomorphism taking (π, y) to ((1, 2)^y, 0), which takes the center, ${\displaystyle 1\times \mathbb {Z} /2\mathbb {Z} }$, 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 ⇐ Strictly characteristic subgroup ⇐ Fully characteristic subgroup ⇐ Verbal subgroup

Examples

Finite example

Consider the group G = S₃ × ${\displaystyle \mathbb {Z} _{2}}$ (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 isomorphic to its second factor ${\displaystyle \mathbb {Z} _{2}}$. Note that the first factor, S₃, contains subgroups isomorphic to ${\displaystyle \mathbb {Z} _{2}}$, for instance {e, (12)} ; let ${\displaystyle f:\mathbb {Z} _{2}<\rightarrow {\text{S}}_{3}}$ be the morphism mapping ${\displaystyle \mathbb {Z} _{2}}$ onto the indicated subgroup. Then the composition of the projection of G onto its second factor ${\displaystyle \mathbb {Z} _{2}}$, 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, ${\displaystyle \mathbb {Z} _{2}}$, 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

• Characteristically simple group

References

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