Commutator subgroup
is the unique smallest normal subgroup
such that
is abelian.
If is a group, then
is a normal subgroup of
and
is abelian. If
is a normal subgroup of
, then
is abelian iff
contains
.
Proof
Let be any automorphism. Then
It follows that . In particular, if
is the automorphism given by conjugation by
, then
, so
.
Since ,
and hence
is abelian.
() If
is abelian, then
for all
. Hence
. Therefore,
contains all commutators and
.
() If
, then
. Thus
for all
. Hence
is abelian.
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