Centers of symmetric groups So I've been working through Beachy/Blairs Abstract Algebra book, and on the last few sections I seem to get continually hung up on questions dealing with centralizers.

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*The last one I encountered was, "Show that if $ n \ge3 $, then then center of $ S_n $ is trivial."
I was able to do this by contradiction and by exploiting something I knew about permutation groups, but for general groups I have an absolutely horrid time working them:


*Let G be a group and let $a \in G $. Show $C(a)$ is a subgroup of G and show $\langle a \rangle \subseteq C(a) $.
What are some good things to think about when I go to solve problems like these?
Thanks,

@Andrew W., observe that for a group $G$ the center $Z(G) =  \bigcap_{g \in G} C_G(g)$. Hence if you start calculating some of the centralizers, you will get a hint of which elements are in the center and which elements are not.
