Lower central series
The lower central series of a group is a particular decreasing sequence of subgroups of that group.
Specifically, let be a group. The lower central series of
is the sequence
defined recursively as follows:
where
denotes the commutator group of two subgroups
of
. It follows from induction that
is a subgroup of
.
A group is called nilpotent if
is the trivial group for sufficiently large
.
Theorem 1. Let and
be groups, and let
be a group homomorphism mapping
into
. Then for all positive integers
,
Thus when
is surjective,
. Also, the subgroup
is characteristic (and in particular, normal) in
.
Proof. We induct on to prove the main statement. For
, we have
and the theorem follows.
Now suppose the theorem holds for . Since the group
is generated by the elements of the form
, for
and
, it follows that
. Since
and
, it follows similarly that
; equality evidently occurs when
is surjective. By applying the theorem to the automorphisms of
, we see that
is a characteristic subgroup of
.
Theorem 2. For all positive integers ,
.
Proof. We use strong induction on the quantity . Our base cases,
and
, follow from definition.
Now, suppose that , and that the inductive hypothesis holds. Then by properties of commutators,
By inductive hypothesis,
, so
Also by inductive hypothesis,
Hence
as desired.