Commutator of two subgroups
The commutator of two subgroups of a group is defined as the subgroup generated by commutators between elements in the two subgroups.
Definition with symbols
is the commutator of the elements and .
Note that there are two conventions for commutators; in some other conventions:
Whatever the convention, the set of commutators is the same; the commutator of and in the former convention equals the commutator of and in the latter convention.
Commutator, closure and join
If are subgroups, let denote the closure of under the action of . Define analogously. We then have:
- is a normal subgroup inside . In fact, , where normalizes .
- is a normal subgroup inside . In fact, where normalizes .
- is a normal subgroup inside . Both and are normal inside , with .
For full proof, refer: Commutator of two subgroups is normal in join
Normalizing characterized in terms of commutators
For subgroups , is contained in the normalizer of if and only if . (In particular, is normal if and only if ).
Similarly, is contained in the normalizer of if and only if . Thus, the subgroups and normalize each other iff . In particular, if both subgroups are normal, their commutator is contained in their intersection.
Permuting subgroups characterized in terms of commutators
Subgroups are permuting subgroups if and only if ; in other words, the commutator of the subgroups is contained in their product.
Normal closure and quotient
The commutator of two subgroups need not, in general, be a normal subgroup. The normal closure of the commutator of two subgroups is of greater interest. If denotes the normal closure of for subgroups of , then the images of and in commute element-wise. Conversely, any normal subgroup for which the images of and commute element-wise in the quotient, must be contained in .
However, in the special case when both and are normal, the commutator of the subgroups is also normal. Further information: Commutator of normal subgroups is normal