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).