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Problem 61 (The Center Of A Dihedral Group)

Let $G=D_4$, the automorphism group of the square. Recall that $Z(G)$ is the center of the group, or the set of elements that commute with every element of the group.

  1. What is $\langle R_{90} \rangle$? What is $\langle R_{180} \rangle$? What is $\langle R_{270} \rangle$? What is $\langle H \rangle$?
  2. Does $R_{90}\in Z(G)$? Explain. (Does $R_{90}$ commute with every element in $G$? In particular, does $R_{90}H=HR_{90}$?)
  3. Compute the center $Z(G)$ and show that it consists of more than just $R_0$. Make sure you can explain why each element is either in $Z(G)$, or not in $Z(G)$.

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