Assignment:
Q1. Define (C_G)(H) = {g is a number in G: g h = h g for all h is a number in H), where H is a subgroup of the group G. Prove that (C_G)(H) is a subgroup of G. Note: (C_G)(H) is called the centralizer of H in G.
Q2. Define (N_G)(H) = {g is a number in G: gH = Hg], where H is a subgroup of the Group G. Prove that (N_G)(H) is a subgroup of G. Note: (N_G)(H) is called the normalizer of H in G.
Q3. Prove that (C_G)(H) is a normal subgroup of (N_G)(H), where H is a subgroup fo the group G.
Q4. Let G be a subgroup such that |G| = p q, where p and q are primes and p < q. Prove that G must have a normal subgroup.
Provide complete and step by step solution for the question and show calculations and use formulas.