Harman Patil (Editor)

Characteristic subgroup

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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.

Contents

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, 2 × 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 GH char G.

    Moreover, while normality is not transitive, it is true that every characteristic subgroup of a normal subgroup is normal.

    H char KGHG

    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 < GH 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 2. Note that the first factor, S3, contains subgroups isomorphic to 2, for instance {e, (12)} ; let f: ℤ2 → S3 be the morphism mapping 2 onto the indicated subgroup. Then the composition of the projection of G onto its second factor 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, 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

    Characteristic subgroup Wikipedia


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