# 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).