Lower central series
From AoPSWiki
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.
