Fully 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. 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 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.

A subgroup of H that is invariant under all inner automorphisms is called normal; also, an invariant subgroup.

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:

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.

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