Let G = $D_n$ and let R be the set of rotations of G.
a. Prove that R < G. What are the left cosets?
b. Suppose n is even and n = 2k. Let H = {$R_0, R_k$}, that is, H is the subgroup consisting of the
identity and the rotation corresponding to a 180° rotation.
i. Prove H < G. (Hint: Prove by cases when x is a rotation or a reflection. When x is
a reflection, use the rules for combining elements of $D_n$ using the subscripts.)
ii. What is the order of $R_1H$ as an element of G/H? (Hint: What is the smallest power
of $R_1$ that is in H?