Problem 1. Let n 3 be an integer. Consider the dihedral group Dn.

a) Find all conjugacy classes of Dn. (Hint: your answer should depend on the parity of n.)

b) Choose a set, R, of representatives of the conjugacy classes of Dn and compute Z(r) for each r R.

c) Prove that Dn = H |x K for subgroups H and K of Dn with K <|/ Dn and |K| = n.

(b): Z(r) is the centralizer of element x

(c): |x is the semi direct product of H and K

<|/ is normal subgroup i.e, K is a normal subgroup of Dn