Deltadesu wrote:The identity is 0, not p, because this is a subgroup of the additive group of integers, and the identity of the additive group of integers is 0. Hopefully that clears up your question.
In particular, the only nontrivial proper subgroups of the additive group of integers (Z,+) are of the form (nZ,+), where n is a positive integer. Eg, (2Z, +) is the group (..., -2, 0, 2, 4, ...) under addition. Clearly, (pZ, +) contains exactly p, pq, and p^q, so you're done.
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