Sorry about the long title. I recently had a few homework problems which were similar to the title of the post. I struggled a bit with them so I'm trying to find additional problems to work on until I can master these ideas... Basically, if we have a group G s.t. |G|=n, and we know G has a conjugacy class C(x) s.t. |C(x)|=m, what can we say about the order of the element x? I know we can work with the center. i.e., since m | n, we have that n=mq for some integer q, which imples that the centralizer of x is s.t. |Z(x)|=q (by the counting formula). Moreover, since Z(x) contains the center of G, Z(G), and since |Z(x)|≠n, x is not an element of the center. Hence, Z(G) has an order of at most the largest integer smaller than q-1 which divides n. Sorry if this is convoluted. I repeat, this isn't a homework problem (anymore) but merely something I'm trying to rectify in my head.