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**[SOLVED] Question about a subset of Z**

So I was working on exercises out of Gallain's Algebra book and it looks like it doesn't actually have an answer! So of course I disagree with the answer in the back of the book, maybe I'm missing something.

**Context**Only provided theory is the definition of a group (chapter 2 in Gallain).

**Problem**

(From the GRE Practice Exam) Let [tex]p[/tex] and [tex]q[/tex] be distinct primes. Suppose that H is a proper subset of the integers and H is a groupunder additionthat contains exactly three elements of the set [tex]\{ p,p+q,pq,p^q,q^p \}[/tex]. Determine which of the following are the three elements in H.

(a) [tex]pq, p^q, q^p[/tex]

(b) [tex]p+q,pq,p^q[/tex]

(c) [tex]p,p+q,pq[/tex]

(d) [tex]p,p^q,q^p[/tex]

(e) [tex]p,pq,p^q[/tex]

**My work**

The identity for the addition operator is 0. If H is a group it must contain the identify as one of the elements. That implies that one of those four elements is 0.

(a) It's not p because p is prime, and 0 is not prime by definition.

(b) Similarly, it's not q.

(c) If [tex]p^q = 0 \Rightarrow p = 0[/tex] but p is not 0, so it's not [tex]p^q[/tex].

(d) Similarly, it's not [tex]q^p[/tex].

(e) If [tex]p + q = 0[/tex] then either p or q is negative. Negative integers are not prime by definition, so it's not p+q.

I have just shown that none of the elements are the identity, and so H is not a group and the problem is ill-posed.

**Correct Solution (back of the book)**(e)

Where have I gone wrong? Before you ask, I didn't type it wrong, I reproduced it exactly as it appeared in the book.