Commutator of two subgroups

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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] := \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.

Facts

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.

For full proof, refer: Commutator of two subgroups is normal in join

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