---
tag: 027ZQS04SJ
kind: statement
role: proposition
name: Quotients of solvable groups are solvable
msc: [20F16]
uses: [0P3KZK0FTV, 0R3H35G96G, 0YYWCNTPWR]
sources:
  - url: https://en.wikipedia.org/wiki/Solvable_group
    locator: Properties
---

## Statement

Let $G$ be a [[solvable group|0FXJ7YFMMX]] and let $\varphi \colon G \to K$ be a surjective [[group homomorphism|0JYTG5K9Y4]].
Then $K$ is solvable.
In particular, $G/N$ is solvable for each [[normal subgroup|0CB0DX5BVV]] $N$ of $G$.

## Remarks

A proof uses the [[derived series|0Q5G5DJ4P2]]: $\varphi(G^{(i)}) = K^{(i)}$ for each $i$, because $\varphi$ maps commutators onto commutators.
So $G^{(n)} = \{e\}$ implies $K^{(n)} = \{e\}$.
The second part follows with the projection $G \to G/N$.

## Explanations

### L1 human

A surjective homomorphism sends the commutators of $G$ onto the commutators of $K$.
So the derived series of $K$ is the image of the derived series of $G$.
If one reaches the identity, the image does too.