# 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.