The Alternating group is generated by its -cycles. That is, every element of can be made by multiplying together -cycles only.
Proof
The product of two transpositions is a product of -cycles:
- .
Therefore any permutation which is a product of evenly-many transpositions (that is, all of ) is a product of -cycles, because we can group up successive pairs of transpositions.
Conversely, every -cycle is in because .