There is a Hungerford problem to prove that the set of integers cannot be written as direct product of any family of its proper subgroups. In such questions, suppose if I move by the way of contradiction and take Z to be the direct product of its two proper subgroups, can I use that intersection of the proper subgroups should be trivial. My main doubt is: In such questions, one can think of two ways: internal direct product or the group is isomorphic to external direct product of its subgroups. Is it legitimate to use the first way as the question is not mentioning it ?
My second question is: are the two ways equivalent or in other words: if a group is isomorphic to external direct product of its subgroups, is it necessary that it is the internal weak direct product of some of its subgroups?