Characteristic subgroup

A characteristic subgroup is a subgroup preserved by every automorphism of G, not just by inner automorphisms (conjugations). It is the notion of "canonical" that matters for anything defined purely from the group structure — every invariant subgroup you can write down without naming elements (center, commutator, Frattini, Fitting, …) is characteristic.

Definition

An automorphism of G is an isomorphism φ : G → G. They form a group Aut(G) under composition.

A subgroup H ≤ G is characteristic in G, written H char G, if

φ(H) = H   for every  φ ∈ Aut(G).

As with normality, φ(H) ⊆ H for all φ is enough — applying the condition to φ⁻¹ gives the reverse inclusion.

Relation to normal subgroups

Every inner automorphism x ↦ gxg⁻¹ is an automorphism, so

H char G   ⟹   H ⊴ G.

The converse fails: a subgroup can be stable under all conjugations without being stable under all automorphisms.

Canonical counterexample — Klein four-group. In V = C₂ × C₂, the three order-2 subgroups

⟨(1, 0)⟩,   ⟨(0, 1)⟩,   ⟨(1, 1)⟩

are all normal (V is abelian, so every subgroup is normal). But Aut(V) ≅ S₃ acts on the three non-identity elements by permutation, so it permutes the three order-2 subgroups transitively. No single one is characteristic; each is a conjugate (under Aut(V)) of the others.

Examples

Each of these is characteristic because its definition uses only the group operation, so it must be preserved by any automorphism:

Center Z(G) — elements commuting with everything.
Commutator subgroup [G, G] — generated by commutators.
Derived series G⁽ⁿ⁾, lower central series γₙ(G), upper central series Zₙ(G) — iterated constructions from commutators and centers.
Frattini subgroup Φ(G) — intersection of maximal subgroups.
Fitting subgroup F(G) — largest normal nilpotent subgroup.
Verbal subgroups — generated by all values of a word, e.g. ⟨xⁿ : x ∈ G⟩, the n-th power subgroup.
Unique subgroup of a given order. If G has exactly one subgroup of order m, it is characteristic.

Properties

Characteristic ⟹ normal.
Closure. Arbitrary intersections and joins of characteristic subgroups are characteristic.
Transitive. K char H and H char G imply K char G. (Contrast normality, which is not transitive.)
Characteristic-in-normal is normal. K char H ⊴ G implies K ⊴ G. This is the usual way the derived series of a normal subgroup — or the terms of the lower central series — is shown to be normal in the ambient group.
Fixed under Aut(G). Aut(G) acts on the set of characteristic subgroups (trivially — it fixes each one). The point is that it acts on every subgroup, and the characteristic ones are the fixed points of that action.

Aut(G) along a characteristic series

Every characteristic subgroup H char G gives a restriction homomorphism

Aut(G)  ⟶  Aut(H) × Aut(G/H),
    φ    ↦  (φ|_H,   φ on G/H).

The image lands inside the pairs of automorphisms that are compatible — they agree on how H sits inside G. This decomposes Aut(G) along any characteristic filtration

1 = H₀ ⊴ H₁ ⊴ H₂ ⊴ ⋯ ⊴ Hₙ = G,      each Hᵢ char G,