Commutator (group)
In a group, the commutator of two elements and
, denoted
or
, is the element
. If
and
commute, then
. More generally,
, or
It then follows that
We also have
where
denote the image of
under the inner automorphism
, as usual.
Relations with Commutators
Proposition. For all in a group, the following relations hold:
;
;
;
;
.
Proof. For the first equation, we note that
From the earlier relations,
hence the relation. The second equation follows from the first by passing to inverses.
For the third equation, we define . We then note that
By cyclic permutation of variables, we thus find
For the fourth equation, we have
The fifth follows similarly.
Commutators and Subgroups
If and
are subgroups of a group
,
denotes the subgroup generated by the set of commutators of the form
, for
and
.
The group is trivial if and only if
centralizes
. Also,
if and only if
normalizes
. If
and
are both normal (or characteristic), then so is
, for if
is an (inner) automorphism, then