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 : GG

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 : GG 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

See also: characteristically simple Untergruppe


  • Art and Cultures
    • Art (
    • Architecture (
    • Cultures (
    • Music (
    • Musical Instruments (
  • Biographies (
  • Clipart (
  • Geography (
    • Countries of the World (
    • Maps (
    • Flags (
    • Continents (
  • History (
    • Ancient Civilizations (
    • Industrial Revolution (
    • Middle Ages (
    • Prehistory (
    • Renaissance (
    • Timelines (
    • United States (
    • Wars (
    • World History (
  • Human Body (
  • Mathematics (
  • Reference (
  • Science (
    • Animals (
    • Aviation (
    • Dinosaurs (
    • Earth (
    • Inventions (
    • Physical Science (
    • Plants (
    • Scientists (
  • Social Studies (
    • Anthropology (
    • Economics (
    • Government (
    • Religion (
    • Holidays (
  • Space and Astronomy
    • Solar System (
    • Planets (
  • Sports (
  • Timelines (
  • Weather (
  • US States (


  • Home Page (
  • Contact Us (

  • Clip Art (
Personal tools