Assignment:
Let G be a group with A and B subgroups of G. Prove that the set AB = {ab | a is in A and b is in B} is a subgroup of G if and only if AB = BA ( ie, for any a in A, b in B, there exist elements a1 in A, b_1 in B such that ab = b_1a_1, and there exist elements a_2 in A, b_2 in B such that ba = a_2b_2)
Provide complete and step by step solution for the question and show calculations and use formulas.