# Alternating group is generated by its three-cycles

The alternating group $$A_n$$ is generated by its $$3$$-cycles. That is, every element of $$A_n$$ can be made by multiplying together $$3$$-cycles only.

# Proof

The product of two transpositions is a product of $$3$$-cycles:

• $$(ij)(kl) = (ijk)(jkl)$$

• $$(ij)(jk) = (ijk)$$

• $$(ij)(ij) = e$$.

Therefore any permutation which is a product of evenly-many transpositions (that is, all of $$A_n$$) is a product of $$3$$-cycles, because we can group up successive pairs of transpositions.

Conversely, every $$3$$-cycle is in $$A_n$$ because $$(ijk) = (ij)(jk)$$.

Parents:

• Alternating group

The alternating group is the only normal subgroup of the symmetric group (on five or more generators).