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.[1] [2] 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 of a group is called a characteristic subgroup if for every automorphism of, one has ; then write .
It would be equivalent to require the stronger condition = for every automorphism of, because implies the reverse inclusion .
Given, every automorphism of induces an automorphism of the quotient group, which yields a homomorphism .
If has a unique subgroup of a given index, then is characteristic in .
See main article: Normal subgroup. A subgroup of that is invariant under all inner automorphisms is called normal; also, an invariant subgroup.
Since 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:
. Serge Lang . Algebra . . . 2002 . 0-387-95385-X.