(4) (4 x 10 = 40 points) Suppose that H, K are distinct nontrivial proper subgroups of a group G such that r-'HICH and r-'KICK for all 1 E G. (a) Is the set HK a subgroup of G? If yes, prove that it is really a subgroup. If no, explain why. (b) Prove that

(4) (4 x 10 = 40 points) Suppose that H, K are distinct nontrivial proper subgroups of a group G such that r-'HICH and r-'KICK for all 1 E G. (a) Is the set HK a subgroup of G? If yes, prove that it is really a subgroup. If no, explain why. (b) Prove that r-'(HK). CHK for all I EG. (c) Is it true that r-'(HK)x= HK for all & E G. If yes, prove the equality. If no, give a counterexample. (a) Determine a condition to guarantee that the elements of H and K commute, and prove your claim.