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.
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.
Definition
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 .
Basic properties
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 .
Related concepts
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:
Let  be a nontrivial group, and let  be the direct product, .
Then the subgroups,  and , are both normal, but neither is characteristic.
In particular, neither of these subgroups is invariant under the automorphism, , that switches the two factors.
For a concrete example of this, let  be the Klein four-group (which is isomorphic to the direct product, ).
Since this group is abelian, every subgroup is normal; but every permutation of the 3 non-identity elements is an automorphism of , so the 3 subgroups of order 2 are not characteristic.
Here .
Consider  and consider the automorphism, ; then  is not contained in .
In the quaternion group of order 8, each of the cyclic subgroups of order 4 is normal, but none of these are characteristic.
However, the subgroup, , is characteristic, since it is the only subgroup of order 2.
If  is even, the dihedral group of order  has 3 subgroups of index 2, all of which are normal.
One of these is the cyclic subgroup, which is characteristic.
The other two subgroups are dihedral; these are permuted by an outer automorphism of the parent group, and are therefore not characteristic.
Strictly characteristic subgroup{{anchor|Strictly invariant subgroup}}
A , or a , which is invariant under surjective endomorphisms.
For finite groups, surjectivity of an endomorphism implies injectivity, so a surjective endomorphism is an automorphism; thus being strictly characteristic is equivalent to characteristic.
This is not the case anymore for infinite groups.
Fully characteristic subgroup{{anchor|Fully invariant subgroup}}
For an even stronger constraint, a fully characteristic subgroup (also, fully invariant subgroup; cf. invariant subgroup), , of a group , is a group remaining invariant under every endomorphism of ; that is,
.
Every group has itself (the improper subgroup) and the trivial subgroup as two of its fully characteristic subgroups.
The commutator subgroup of a group is always a fully characteristic subgroup.
Every endomorphism of  induces an endomorphism of , which yields a map .
Verbal subgroup
An even stronger constraint is verbal subgroup, which is the image of a fully invariant subgroup of a free group under a homomorphism.
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.
Transitivity
The property of being characteristic or fully characteristic is transitive; if  is a (fully) characteristic subgroup of , and  is a (fully) characteristic subgroup of , then  is a (fully) characteristic subgroup of .
.
Moreover, while normality is not transitive, it is true that every characteristic subgroup of a normal subgroup is normal.
Similarly, while being strictly characteristic (distinguished) is not transitive, it is true that every fully characteristic subgroup of a strictly characteristic subgroup is strictly characteristic.
However, unlike normality, if  and  is a subgroup of  containing , then in general  is not necessarily characteristic in .
Containments
Every subgroup that is fully characteristic is certainly strictly characteristic and characteristic; but a characteristic or even strictly characteristic subgroup need not be fully characteristic.
The center of a group is always a strictly characteristic subgroup, but it is not always fully characteristic.
For example, the finite group of order 12, , has a homomorphism taking  to , which takes the center, , into a subgroup of , which meets the center only in the identity.
The relationship amongst these subgroup properties can be expressed as:
Subgroup ⇐ Normal subgroup ⇐ Characteristic subgroup ⇐ Strictly characteristic subgroup ⇐ Fully characteristic subgroup ⇐ Verbal subgroup
Examples
Finite example
Consider the group  (the group of order 12 that is the direct product of the symmetric group of order 6 and a cyclic group of order 2).
The center of  is isomorphic to its second factor .
Note that the first factor, , contains subgroups isomorphic to , for instance ; let  be the morphism mapping  onto the indicated subgroup.
Then the composition of the projection of  onto its second factor , followed by , followed by the inclusion of  into  as its first factor, provides an endomorphism of  under which the image of the center, , is not contained in the center, so here the center is not a fully characteristic subgroup of .
Cyclic groups
Every subgroup of a cyclic group is characteristic.
Subgroup functors
The derived subgroup (or commutator subgroup) of a group is a verbal subgroup.
The torsion subgroup of an abelian group is a fully invariant subgroup.
Topological groups
The identity component of a topological group is always a characteristic subgroup.
See also
Characteristically simple group
References
