---
tag: 36FRFTA090
kind: proof
name: Proof that the symmetric group is generated by transpositions
msc: [20B30]
proves: 0AQNX7G443
uses: [0RJMNEN3BJ, 02DTRYKGE8, 0RG9SPQSWM, 0N0PXCTYA7, 0XK6FECMW5]
sources:
  - url: https://en.wikipedia.org/wiki/Symmetric_group
    locator: Generators and relations
---

## Proof

1. Let $n \ge 2$, let $S_n$ be the [[symmetric group|0RJMNEN3BJ]], and let $T$ be the set of all [[transpositions|02DTRYKGE8]] in $S_n$.
2. By definition, the [[subgroup generated by|0RG9SPQSWM]] $T$ is the intersection $\langle T \rangle$ of all subgroups of $S_n$ that contain $T$. In particular, $\langle T \rangle \subseteq S_n$.
3. Let $U$ be a subgroup of $S_n$ with $T \subseteq U$. A [[subgroup|0N0PXCTYA7]] is closed under products, so by induction on the number of factors, $U$ contains each product of elements of $T$.
4. By the [[decomposition into transpositions|0XK6FECMW5]], each $\sigma \in S_n$ is such a product. So $U = S_n$.
5. By steps 3 and 4, the only subgroup of $S_n$ that contains $T$ is $S_n$ itself. Hence $\langle T \rangle = S_n$.

## Idea

Any subgroup that contains all transpositions contains all their products. Every permutation is such a product, so that subgroup is the whole group.