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 characteristic subgroup, $H char G$ , if for every automorphism $φ$ of $G$ , $φ[H] \leq H$ 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 $Aut(G) \to Aut(G / H)$ .

If $G$ has a unique subgroup $H$ of a given (finite) 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[H] \leq H, \forallφ \in Inn(G)

Since $Inn(G) \subseteq Aut(G)$ 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, $H \times H$ . Then the subgroups, ${1} \times H$ and $H \times {1}$ , are both normal, but neither is characteristic. In particular, neither of these subgroups is invariant under the automorphism, $(x, y) \to (y, x)$ , 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, $ℤ 2 \times ℤ 2$ ). 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 $n$ is even, the dihedral group of order $2 n$ 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] \leq H, \forallφ \in 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.^[3]^[4]

Every endomorphism of $G$ induces an endomorphism of $G/H$ , which yields a map $End(G) \to End(G / H)$ .

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 \Rightarrow 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 \Rightarrow 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, $Sym(3) \times ℤ/2ℤ$ , has a homomorphism taking $(π, y)$ to $((1, 2) y, 0)$ which takes the center, $1 \times ℤ/2ℤ$ , into a subgroup of $Sym(3) \times 1$ , 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 $G = S 3 \times ℤ 2$ (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 ${e, (12)}$ ; let $f : ℤ 2 \to S 3$ 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.

References

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

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