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mahler1
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Homework Statement
Let ##G## be a group such that its center ##Z(G)## has finite index. Prove that every conjugacy class has finite elements.
Homework Equations
The Attempt at a Solution
I know that ##[G:Z(G)]<\infty##. If I consider the action on ##G## on itself by conjugation, each conjugacy class is identified with an orbit, and for each orbit ##\mathcal0_x \cong G/C_G(x)##, where ##C_G(x)## is the stabilizer of ##x## by the action, in this particular case, the centralizer of ##x##. I got stuck here, I know that ##Z(G) \leq C_G(x)## for all ##x \in G##, I don't know how to deduce from here that ##[G:C_G(x)]<\infty##, I would appreciate some help.