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 :
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 .