Show that every element of An for n 3 may be expressed as a

Show that every element of A_n for n > 3 may be expressed as a 3-cycle or as a product of 3-cycles.

Solution

This is true for all n if we understand the product of zero 3 -cycles to be the identity. In particular, if n 2, then An = {id}.

To demonstrate the claim, it is enough to show any product (a b)(c d) of two 2 -cycles (with a 6= b, c 6= d) is a product of 3 -cycles. If a, b, c, d are all distinct, then this product is (a c b)(c d a). If exactly three of a, b, c, d are distinct, we can rewrite the transpositions so that a, b, d are distinct and b = c. In this case, the product is just (a b d). If only two of a, b, c, d are distinct, then (after re-writing) the product must be (a b)(a b) = id, which is either a product of zero 3 -cycles (my preference!) or the product of any 3 -cycle with itself three times.