Prove that h c g is a normal subgroup if and only if every left coset is a right coset, i. e., ah = ha for all a e g

Suppose first that is a normal subgroup. Then by definition we must have for all , for every . Let and choose (). By hypothesis we have , i.e. for some , thus . So we have . You can prove in the same way.

Suppose for all . Let , we have to prove   for every . So, let . We have that for some (by the hypothesis). hence we have . Because was chosen arbitrarily  we have the desired .

hi there!

the answer will be d: 2