Assignment:
Let N be a normal subgroup of G, a in G, and C the conjugacy class of a in G.
a) Prove that a in N if and only if C is contained in N.
b) If C_i is any conjugacy class in G, prove that C_i is contained in N or the intersection of C_i and N is empty.
c) Use the class equation to show that |N| = |C_1| + ...+ |C_k|, where C_1, ... , C_k are all the conjugacy classes of G that are contained in N
Provide complete and step by step solution for the question and show calculations and use formulas.