[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 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.