Lower central series
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 .
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.