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Center of a Dihedral Group

  1. Apr 3, 2016 #1

    RJLiberator

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    1. The problem statement, all variables and given/known data
    If n ≥ 3, show that Z(D_n) = C(x) ∩ C(y).

    2. Relevant equations
    G is a group, g∈G
    C(g) = {h∈G: hg = gh } The Centralizer of g
    Z(G) = {h∈G: hg = gh for all g∈G} The center of G

    ∩ means the set of all points that fall in C(x) and C(y).

    Every element of D_n can be uniquely written in the form y^i x^j.
    x is a reflection and y is a rotation

    3. The attempt at a solution

    The elements of Dn are 1, y, y^2, y^2,...y^(n-1) and x, yx, y^2x, ..., y^(n-1)x
    C(x) = { h∈G: hx = xh }
    C(y) = { h∈G: hy = yh }
    Z(D_n) = {h∈G: hx=xh for all x∈G}

    Note: xy^i = y^(n-i)x for 1 ≤ i ≤ n-1

    Okay. I need some help.

    I don't understand how to use the definitions of C(x) and C(y) and Z(D_n) here.
     
  2. jcsd
  3. Apr 3, 2016 #2

    andrewkirk

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    To prove equality, you need to prove inclusion in both directions:
    1. Prove that ##Z(D_n)\subseteq C(x)\cap C(y)##
    2. Prove that ##C(x)\cap C(y)\subseteq Z(D_n)##

    The first one is easier, so I suggest you do that first. Pick an element ##a\in Z(D_n)##. Is it in ##C(x)## (look at the definitions)?. Then see if it's in ##C(y)##.
     
  4. Apr 4, 2016 #3

    RJLiberator

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    My problem is, I don't understand the definitions...

    C(g) = {h∈G: hg = gh } The Centralizer of g
    Z(G) = {h∈G: hg = gh for all g∈G} The center of G

    If we have a existing in Z(Dn) then a is clearly abelian, but then we need to show that it either exists in C(x) or C(y).
    C(y) says hy=yh and C(x) says hx=xh
    If a = xg=gx then we just need to prove that g exists in G?
     
  5. Apr 4, 2016 #4

    andrewkirk

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    C(g) is the set of all elements of G that commute with element g.
    Z(G) is the set of all elements of G that commute with everything.
    Remember that x and y were given specific meanings in the above definition of the dihedral group. They are the names of specific group elements, not just variable names.
     
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