---
tag: Q43237D4KK
kind: proof
name: Proof that nilpotent groups are solvable
msc: [20F18, 20F16]
proves: 0GNG2KVPZJ
uses: [0DZPD2MG33, 0S7Y9Q990Q, 0W77CVBCHQ, 0YYWCNTPWR, 096HN851YZ, 0P3KZK0FTV]
sources:
  - url: https://en.wikipedia.org/wiki/Nilpotent_group
    locator: Properties
---

## Proof

1. Since $G$ is [[nilpotent|0DZPD2MG33]], it has a central series $\{e\} = N_0 \subseteq N_1 \subseteq \cdots \subseteq N_n = G$. Each $N_i$ is a [[normal subgroup|0W77CVBCHQ]] of $G$, and $N_{i+1}/N_i \subseteq Z(G/N_i)$ for $i < n$.
2. Fix $i < n$. Since $N_i$ is normal in $G$, it is normal in the subgroup $N_{i+1}$. So $\{e\} = N_0 \trianglelefteq N_1 \trianglelefteq \cdots \trianglelefteq N_n = G$ is a [[subnormal series|096HN851YZ]].
3. The factor $N_{i+1}/N_i$ is the set of cosets $xN_i$ with $x \in N_{i+1}$. Its product $(xN_i)(yN_i) = xyN_i$ is the product of the [[quotient group|0YYWCNTPWR]] $G/N_i$.
4. By step 1, each such coset lies in the [[centre|0S7Y9Q990Q]] $Z(G/N_i)$. So for $x, y \in N_{i+1}$, we get $(xN_i)(yN_i) = (yN_i)(xN_i)$.
5. So each factor $N_{i+1}/N_i$ is abelian. By step 2, $G$ is [[solvable|0P3KZK0FTV]].

## Idea

A central series is already a subnormal series. Each of its factors sits inside the centre of a quotient group, and elements of a centre commute with each other. So every factor is abelian.