---
tag: 05Y0F9M0TS
kind: statement
role: proposition
name: Subgroups of solvable groups are solvable
msc: [20F16]
uses: [0P3KZK0FTV, 0R3H35G96G, 0N0PXCTYA7]
sources:
  - url: https://en.wikipedia.org/wiki/Solvable_group
    locator: Properties
---

## Statement

Let $G$ be a [[solvable group|0FXJ7YFMMX]] and let $H$ be a [[subgroup|0X1JXT47GA]] of $G$.
Then $H$ is solvable.

## Remarks

A proof uses the [[derived series|0Q5G5DJ4P2]]: $H^{(i)} \subseteq G^{(i)}$ for each $i$, so $G^{(n)} = \{e\}$ implies $H^{(n)} = \{e\}$.
More generally, if there is an injective homomorphism from $H$ into a solvable group, then $H$ is solvable.
For example, each subgroup of $S_4$ is solvable.

## Explanations

### L1 human

Commutators of elements of $H$ are also commutators in $G$.
So the derived series of $H$ stays inside the derived series of $G$.
If the larger series reaches the identity, the smaller one does too.