---
tag: 0KFZ0QX83G
kind: statement
role: proposition
name: The alternating group is generated by three-cycles
msc: [20B30]
uses: [0T13HM377F, 0RG9SPQSWM, 0V0E04DWX2]
sources:
  - url: https://en.wikipedia.org/wiki/Alternating_group
    locator: Generators and relations
---

## Statement

Let $n \geq 3$.
The [[alternating group|0Z5905Q73D]] $A_n$ is [[generated|0RP7VWQ0YN]] by the $3$-cycles in $S_n$.

## Remarks

Each $3$-cycle is even, so the $3$-cycles lie in $A_n$.
An even permutation is a product of an even number of transpositions, and each product of two transpositions is a product of $3$-cycles.
For distinct $a, b, c, d$, the formulas are $(a\ b)(a\ c) = (a\ c\ b)$ and $(a\ b)(c\ d) = (a\ c\ b)(a\ c\ d)$.

## Explanations

### L1 human

An even permutation is a chain of swaps, taken two at a time.
Each pair of swaps can be written with $3$-cycles.
So the $3$-cycles are enough to build all even permutations.