Alternating group is generated by its three-cycles
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 \((ijk) = (ij)(jk)\).
- Alternating group
The alternating group is the only normal subgroup of the symmetric group (on five or more generators).