- #1
RJLiberator
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Homework Statement
If n ≥ 3, show that Z(D_n) = C(x) ∩ C(y).
Homework 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
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.