Commutator of two subgroups
Contents
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 is a group and
and
are subgroups of
. The commutator of the subgroups
and
, denoted
, is defined as:
where:
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.
Facts
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