What is the center of a symmetric group?

Question

How can you show that the center of a Symmetric group (Sn) consists of only the identity, given n>2.

Does the fact that any permutation can be represented as the product of 2-cycles play a part here? I'm truly stuck! Please help! This is a problem from modern/abstract algebra.

Answer 1

Suppose z is in Sn and that z is not the identity. Then z must take some element a to some different element b. Let c be any element of {1, ..., n} different from both a and b (we can find one since n > 2).
Then the cycle g = (b c) does not commute with z, since gz(a) = g(b) = c but zg(a) = z(a) = b.
Hence no non-identity element of Sn can be in the centre of Sn, so the centre of Sn is just the identity.