Let H and K be subgroups of a group G. Ask you to establish necessary and sufficient criteria for G to appear as the internal direct product of H and K. Let H and K be groups and let G = H x K. Recall that both H and K appear as subgroups of G in a natural way. Show that these subgroups H (actually H x {e}) and K (actually {e} x K) have the following properties. 

a. Every element of G is of the form hk for some h ∈ H and k ∈ K. 

b. hk = kh for all h ∈ H and k ∈ K. 

c. H ∩ K = {e}.