## Ascending Central Series of $G$

Let $G$ be a group. The center $C(G)$ of $G$ is a normal subgroup. Let $C_2(G)$ be the inverse image of $C(G/C(G))$ under the canonical projection $G\to G/C(G)$. By Correspondence Theorem, $C_2(G)$ is normal in $G$ and contains $C(G)$.

Continue this process by defining inductively: $C_1(G)=C(G)$ and $C_i(G)$ is the inverse image of $C(G/C_{i-1}(G))$ under the canonical projection $G\to G/C_{i-1}(G)$.

Thus we obtain a sequence of normal subgroups of $G$, called the ascending central series of $G$: $\displaystyle \langle e\rangle

## Nilpotent Group

A group $G$ is nilpotent if $C_n(G)=G$ for some $n$.

## Abelian Group is Nilpotent

Every abelian group $G$ is nilpotent since $G=C(G)=C_1(G)$.

## Every finite $p$-group is nilpotent (Proof)

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

Hence if $G\neq C_i(G)$, then $G/C_i(G)$ is a $p$-group, and $C(G/C_i(G))$ is non-trivial. Thus $C_{i+1}(G)$, the inverse image of $C(G/C_i(G))$ under $\pi:G\to G/C_i(G)$, strictly contains $C_i(G)$.

Since $G$ is finite, $C_n(G)$ must be $G$ for some $n$.