A characteristic subgroup of a group is a subgroup of that is stable under every automorphism on . Since the map is an automorphism (specifically, an inner automorphism) on , for every , it follows that every characteristic subgroup of is also a normal subgroup of .
Every group is a characteristic subgroup of itself; a group's trivial subgroup is characteristic.
Let be a natural number that divides the order of . Then the set of elements of for which divides is a characteristic subgroup of . Since every subgroup of a cyclic group is cyclic, it follows that every subgroup of a cyclic group is a characteristic group.
In general, though, not every cyclic group of an Abelian group is characteristic. For instance, the Klein 4-group has no non-trivial characteristic subgroups, since any permutation of its non-identity elements is an automorphism. For an odd prime , the group has no nontrivial characteristic subgroups either. Indeed, for any relatively prime to , the mapping is an automorphism, as is the mapping . Thus if () is a member of a characteristic subgroup, then so is , and these two evidently generate ; and if () is an element of a characteristic subroup, then setting , we see that is an element of this characteristic subgroup; therefore so is all of .
This idea also shows that has no non-trivial characteristic subgroups, for any natural number . In fact, the characteristic subgroups of are of the form , for some integer .
Characteristic Subgroups of Normal Subgroups
Theorem 1. Let be a group, and let be a normal subgroup of . Let be a characteristic subgroup of . Then is a normal subgroup of ; furthermore, if is a characteristic subgroup of , then so is .
Proof. Let be an (inner) automorphism on . Then its restriction to is an automorphism on , so is characteristic (normal) in if and only if is.
Theorem 2. An equivalence relation is compatible with the group law on and every automorphism on if and only if is equivalent to , for some characteristic subgroup of .
Proof. Evidently, must be of the form , where is the set of elements equivalent to under . For any and any automorphism on , , so is a characteristic subgroup.
On the other hand, if is a characteristic subgroup, the relation is compatible with the group law; it also implies .