Commutator of two subgroups

Definition

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:

$[H,K]:=⟨[h,k]\mid h\in H,k\in K⟩$

where:

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

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

Facts

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. Further information: Commutator of normal subgroups is normal