Can someone elaborate more on this question?

(there is a brief explanation on viewtopic.php?f=1&t=333&p=3839&hilit=9367+55#p3839)

Let p and q be distinct primes. There is a proper subgroup J of the additive groups of integers which contains exactly three elements of the set {p,p+q, pq,p^q, q^p}. Which three elements are in J?

The answer is E.

However, from my basic knowledge of subgroups, the properties of closure, associativity (follows from group), identity and inverse must be fulfilled.

However, if we assume p is the identity, then how do the closure and inverse properties hold?

Thanks for clearing my misunderstanding.