Proof

Every element of A_n is a product of an even number of transpositions. Hence it suffices to prove that each product of two transpositions, different from the identity element, can be written as a product of 3-cycles.

Let (a,b) and (c,d) be two different transpositions.

• If a, b, c and d are pairwise distinct, then
  (a,b)(c,d) = (a,b)(b,c)(b,c)(c,d) = (a,b,c)(b,c,d).

• Without loss of generality we are left with the case where a, b, d are pairwise distinct and b = c. But then
  (a,b)(b,d) = (a,b,d).

This proves the theorem.