For a fixed element a of a group G, the set Ca = {x E G | ax = xa } is the centralizer of a in G.
Prove that for any a E G, Ca is a subgroup of G.
Where is it you are stuck on this question?
Let . Then you need to check that and . This is precisely what you would always do to check something is a subgroup.
Checking that is in is almost elementary (substitute in the above condition for ), but the fact that is not immediately obvious. However, just take the condition, and pre- and post-multiply by to get that the condition also holds for