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.
For finite groups, surjectivity of an endomorphism implies injectivity, so a surjective endomorphism is an automorphism; thus being strictly characteristic is equivalent to characteristic.
More generally, any verbal subgroup is always fully characteristic.
For any reduced free group, and, in particular, for any free group, the converse also holds: every fully characteristic subgroup is verbal.
For example, the finite group of order 12, Sym(3) ×
, into a subgroup of Sym(3) × 1, which meets the center only in the identity.
The relationship amongst these subgroup properties can be expressed as: Consider the group G = S3 ×
Note that the first factor, S3, contains subgroups isomorphic to
, followed by f, followed by the inclusion of S3 into G as its first factor, provides an endomorphism of G under which the image of the center,
, is not contained in the center, so here the center is not a fully characteristic subgroup of G. Every subgroup of a cyclic group is characteristic.
The identity component of a topological group is always a characteristic subgroup.