Derived group
The derived (sub)group (or commutator (sub)group) of a group is the smallest normal subgroup
of
such that the quotient group
is abelian.
Specifically, let be a group. The group
generated by the set of commutators of
is called the derived group of
. It is also called the commutator group of
, though in general it is distinct from the set of commutators of
. It is a normal subgroup of
—in fact, it is a characteristic subgroup.
Evidently, if is a subgroup of
, then
is a subgroup of
.
Proposition. Let and
be groups, and
be a group homomorphism. Then
; in fact,
, so in particular, if
is surjective,
.
Proof. If is a commutator of
, then
is a commutator of
; thus
. Suppose
are points in the image of
under
; let
be elements of
such that
. Then
. Hence
.
Corollary 1. The derived group is a characteristic subgroup of
. In particular, it is normal.
Corollary 2. The quotient group is commutative. Let
be the canonical homomorphism from
to
. Let
be an abelian group. Then every homomorphism
can be expressed uniquely as
, where
is a homomorphism.
Proof. Note that ; thus
is abelian. If
is a homomorphism, then
. Thus the realtion
is compatible with equivalence mod
.
Corollary 3. Let be a subgroup of
. Then
if and only if
is normal in
and
is commutative.
Proof. If contains
, then
is normal in
, since every subgroup of an abelian group is normal; hence
is normal, and
is isomorphic to
, which is commutative. The converse follows from the previous corollary.
Corollary 4. Let be a generating subset of
. Then
is the normal subgroup generated by the set of commutators of elements of
.
Proof. Let be the normal subgroup generated by the commutators of elements of
. By definition,
. On the other hand, the set
generates the group
; since the elements of
commute,
is abelian, and hence
.