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 $φ(H) \leq H$ ; then write $H char G$ .

It would be equivalent to require the stronger condition $φ(H)$ = $H$ for every automorphism $φ$ of $G$ , because $φ -1 (H) \leq H$ implies the reverse inclusion $H \leq φ(H)$ .

Basic properties

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

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.

\forallφ \in Inn(G)： φ[H] \leq H

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, $\mathbb {Z} _{2}\times \mathbb {Z} _{2}$ ). 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 $2 n$ 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 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 $G$ ; that is,

\forallφ \in End(G)： φ[H] \leq 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 $End(G) \to End(G / H)$ .

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 \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, $\mathbb {Z} /2\mathbb {Z}$ , has a homomorphism taking $(π, y)$ to $((1, 2) y, 0)$ , which takes the center, $1\times \mathbb {Z} /2\mathbb {Z}$ , 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 $\mathbb {Z} _{2}$ (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 $\mathbb {Z} _{2}$ . Note that the first factor, $S 3$ , contains subgroups isomorphic to $\mathbb {Z} _{2}$ , for instance ${e, (12)}$ ; let $f:\mathbb {Z} _{2}<\rightarrow {\text{S}}_{3}$ be the morphism mapping $\mathbb {Z} _{2}$ onto the indicated subgroup. Then the composition of the projection of $G$ onto its second factor $\mathbb {Z} _{2}$ , 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, $\mathbb {Z} _{2}$ , 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.

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.

[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.

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