# Characteristic subgroup

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 .

## Examples

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 .