# 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).