Characteristic subgroup

In group theory, a characteristic subgroup of a group G is a subgroup H, which remains fixed under every automorphism of G. That is, a subgroup H of G is called characteristic if for every automorphism ( bijective group homomorphism from G to G ) holds f that f ( H) is a subset of H.

Every characteristic subgroup is a normal subgroup, because it remains firmly under any particular inner automorphism. Conversely, however, not every normal subgroup characteristic. Consider, for example, the Klein four-group. Any of its subgroups is normal but there is an automorphism of the permuted sub ​​-element 2, so none of the 2 -element sub-groups is typical.

If H is a normal subgroup of the finite group G, and G has no further sub- group of the same order, then H is characteristic, since automorphisms represent subgroups only on order same subgroups.

A related concept is that of a strictly characteristic subgroup ( engl. distinguished subgroup ). Such a subgroup H remains fixed under every epimorphism ( surjective homomorphism from G to G). Note that for an infinite group not every epimorphism must be an automorphism.

An even stronger requirement is that a fully characteristic subgroup ( engl. fully characteristic subgroup or fully invariant subgroup ). Such a subgroup H remains fixed under every endomorphism ( homomorphism from G to G), that is, if f: G → G is a homomorphism, then f (H ) subset of H.

Each fully characteristic subgroup is therefore strictly characteristic, but not vice versa. The center of a group is always strictly characteristic, but eg not fully characteristic of the group D6 × C2 ( the direct product of the dihedral group of order 6 with the cyclic group of order 2).

The commutator subgroup of a group is always fully characteristic in it, as well as the torsion subgroup of an abelian group.

The property to be characteristic or fully characteristic is transitive, that is, H is a (fully ) characteristic subgroup of K and K is a (fully ) characteristic subgroup of G, then also H a (fully ) characteristic subgroup of G.

• Group (mathematics)
• Group Theory