Abstract Algebra?

Question

Can some one explain me how to prove that S5 is abelian

Let a = 1 2 3 4 5
4 5 3 2 1

and

b = 1 2 3 4 5
3 4 1 5 2

Answer 1

But S5 is not abelian. (a∘b)(1) = a(b(1)) = a(3) = 3, but (b∘a)(1) = b(a(1)) = b(4) = 5 ≠ 3, so a∘b ≠ b∘a.

Answer 2

Sorry, Abstract Algebra is not part of our curriculum in 1960's and I have no contact to help us on that.