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Center of a group with finite index

  1. Nov 30, 2014 #1
    1. The problem statement, all variables and given/known data

    Let ##G## be a group such that its center ##Z(G)## has finite index. Prove that every conjugacy class has finite elements.

    2. Relevant equations


    3. 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.
     
  2. jcsd
  3. Dec 1, 2014 #2

    pasmith

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    Homework Helper

    Hint: Show that if [itex]g_1[/itex] and [itex]g_2[/itex] are in the same coset of [itex]Z(G)[/itex] then for all [itex]h \in G[/itex] we have [itex]g_1hg_1^{-1}= g_2hg_2^{-1}[/itex].

    (Edit: the converse also holds, but I don't think you will require that.)

    Given any [itex]h \in G[/itex] you can then show that [itex]\phi_h : G/Z(G) \to [h] : gZ(G) \mapsto ghg^{-1}[/itex] is a surjection, where [itex][h][/itex] is the conjugacy class of [itex]h[/itex]. (The first hint allows us to define such a function.)
     
    Last edited: Dec 1, 2014
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