Commutator of two subgroups

Symbol-free definition
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
Suppose $$G$$ is a group and $$H$$ and $$K$$ are subgroups of $$G$$. The commutator of the subgroups $$H$$ and $$K$$, denoted $$[H,K]$$, is defined as:

$$[H,K] := \langle [h,k] \mid h \in H, k \in K \rangle$$

where:

$$[h,k] = h^{-1}k^{-1}hk$$

is the commutator of the elements $$h$$ and $$k$$.

Note that there are two conventions for commutators; in some other conventions:

$$[h,k] = hkh^{-1}k^{-1}$$.

Whatever the convention, the set of commutators is the same; the commutator of $$h$$ and $$k$$ in the former convention equals the commutator of $$h^{-1}$$ and $$k^{-1}$$ in the latter convention.

Commutator, closure and join
If $$H, K \le G$$ are subgroups, let $$H^K$$ denote the closure of $$H$$ under the action of $$K$$. Define $$K^H$$ analogously. We then have:


 * $$[H,K]$$ is a normal subgroup inside $$H^K$$. In fact, $$H^K = H[H,K]$$, where $$H$$ normalizes $$[H,K]$$.
 * $$[H,K]$$ is a normal subgroup inside $$K^H$$. In fact, $$K^H = K[H,K]$$ where $$K$$ normalizes $$[H,K]$$.
 * $$[H,K]$$ is a normal subgroup inside $$\langle H, K \rangle$$. Both $$H^K$$ and $$K^H$$ are normal inside $$\langle H, K \rangle$$, with $$\langle H, K \rangle = KH^K = HK^H$$.

Normalizing characterized in terms of commutators
For subgroups $$H,K \le G$$, $$K$$ is contained in the normalizer of $$H$$ if and only if $$[H,K] \le H$$. (In particular, $$H$$ is normal if and only if $$[H,G] \le H$$).

Similarly, $$H$$ is contained in the normalizer of $$K$$ if and only if $$[H,K] \le K$$. Thus, the subgroups $$H$$ and $$K$$ normalize each other iff $$[H,K] \le H \cap K$$. In particular, if both subgroups are normal, their commutator is contained in their intersection.

Permuting subgroups characterized in terms of commutators
Subgroups $$H, K \le G$$ are permuting subgroups if and only if $$[H,K] \le HK$$; 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 $$L$$ denotes the normal closure of $$[H,K]$$ for $$H,K$$ subgroups of $$G$$, then the images of $$H$$ and $$K$$ in $$G/L$$ commute element-wise. Conversely, any normal subgroup for which the images of $$H$$ and $$K$$ commute element-wise in the quotient, must be contained in $$L$$.

However, in the special case when both $$H$$ and $$K$$ are normal, the commutator of the subgroups is also normal.