Characteristic subgroup

In abstract algebra, a characteristic subgroup of a group G is a subgroup H of G invariant under each automorphism of G. This means that if f : G -> G is a group automorphism (a bijective homomorphism from the group G to itself), then for every x in H we have f(x) in H.

Characteristic subgroups are in particular invariant under inner automorphisms, so they are normal subgroups. However, the converse is not true; for example, consider the Klein group V₄. Every subgroup of this group is normal; but there is an automorphism which essentially "swaps" the various subgroups of order 2, so these subgroups are not characteristic.

On the other hand, if H is a normal subgroup of G, and there are no other subgroups of the same order, then H must be characteristic; since automorphisms are order-preserving.

A related concept is that of a strictly characteristic subgroup. In this case the subgroup H is invariant under the applications of surjective endomorphisms. (Recall that for an infinite group, a surjective endomorphism is not necessarily an automorphism).

For an even stronger constraint, a fully characteristic subgroup (also called a fully invariant subgroup) H of a group G is a group remaining invariant under every endomorphism of G; in other words, if f : G → G is any homomorphism, then f(H) is a subgroup of H.

Every fully characteristic subgroup is, perforce, a characteristic subgroup; but a characteristic subgroup need not be fully characteristic. The center of a group is always a strictly characteristic subgroup, but not always fully characteristic; for example, consider the group D₆ × C₂ (the direct product of a dihedral group and a cyclic group of order 2).

The derived subgroup (or commutator subgroup) of a group is always a fully characteristic subgroup, as is the torsion subgroup of an abelian group.

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.

The relationship amongst these types of subgroups can be expressed as:

subgroup ← normal subgroup ← characteristic subgroup ← strictly characteristic subgroup ← fully characteristic subgroup