I recently came across an exercise that is easy to do, but I would like to see if there is a faster/more analytical way to do it. Right now in order to find the center of, say, S3 I need to write down the multiplication table and compare the rows and columns for each combination of elements to see if the multiplication is commutative to include it in with the center. I'm thinking there must be a way to calculate the center of a group more analytically? I mean, what if I've got to find the center of S5 for example. The table itself could take a couple of hours to calculate.
I did have a thought. The same exercise had me calculating the centralizer of each element in the group. I haven't yet sat down to prove or disprove this, but it would seem to me that the center of the group would be the intersection of the centralizers of each element?