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.
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?