## Ascending Central Series of

Let be a group. The center of is a normal subgroup. Let be the inverse image of under the canonical projection . By Correspondence Theorem, is normal in and contains .

Continue this process by defining inductively: and is the inverse image of under the canonical projection .

Thus we obtain a sequence of normal subgroups of , called the **ascending central series** of :

## Nilpotent Group

A group is nilpotent if for some .

## Abelian Group is Nilpotent

Every abelian group is nilpotent since .

## Every finite -group is nilpotent (Proof)

and all its nontrivial quotients are -groups, and therefore have non-trivial centers.

Hence if , then is a -group, and is non-trivial. Thus , the inverse image of under , strictly contains .

Since is finite, must be for some .