Alternating group is generated by its three-cycles

https://arbital.com/p/alternating_group_generated_by_three_cycles

by Patrick Stevens Jun 17 2016

A useful result which lets us prove things about the alternating group more easily.


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 .