what you want to prove, is that if α is a n-cycle, with α = (a_{1}a_{2}... a_{n}),

then α^{k}sends a_{j}to a_{j+k (mod n)}(you can do this by induction on k).

now if n is even, say n = 2m, then α^{2}sends:

a_{1}→a_{3}→...→a_{n-1}→a_{1}(since n-1 = 2m -1 is odd)

a_{2}→a_{4}→...→a_{n}→a_{2}

that is α^{2}= (a_{1}a_{3}... a_{n-1})(a_{2}a_{4}... a_{n}),

so α splits into 2 disjoint m-cycles.

but if n is odd, say n = 2m+1, so that n-2 is odd, then:

α^{2}= (a_{1}a_{3}.... a_{n-2}a_{n}a_{2}a_{4}... a_{n-1}),

which is an n-cycle.