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Concept
Characteristic subgroup
Summary
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.
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).
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.
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, ). 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.
Official source
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