If p and q are prime which elements are in the subgroup? (GRE question)

Question

I was just doing some practice problems in my abstract algebra book trying to get a warm up this morning, but I found a GRE problem in the problem set and I don't know how to solve it. I've tried to think of examples of proper subgroups of integers under addition so I might get an intuition, but I was able to.

Let $p$ and $q$ be distinct primes. Suppose that $H$ is a proper subset of the integers that is a group under addition that contains exactly three elements of the set $\{p,p+q,pq,p^{q},q^{p}\}$. Determine which of the following three elements are in $H$.

The answer is $p,pq,p^{q}$. However, I am completely confused as how to get here. I thought I might know something, but I don't really have any reasons for why they might be, so I figured I don't actually know why.

Could someone show me the proof or reasoning why this is so?

Answer 1

The additive subgroups of ${\mathbb Z}$ are all of the form $\{kn: k \in {\mathbb Z}\}$ for an integer $n$. In other words $H$ consists of all multiples of some integer $n$. Here $n$ can't be $1$ or $-1$ since it is a proper subgroup.

The only subset of the $5$-member set that consists of multiples of a single integer $n \neq 1$ or $-1$ are $\{p,pq,p^q\}$, which are all multiples of $p$. Hence that's your answer.

Answer 2

Let $K$ be a subgroup of $\mathbb{Z}$. If there are elements $a,b\in K$ such that $\gcd(a,b)=1$, then by Bezout's identity (Wikipedia link) we know $1\in K$, hence $n\in K$ for every $n\in \mathbb{Z}$, i.e. $K=\mathbb{Z}$.

Because $H$ is required to be a proper subgroup, there cannot be any two such elements in $H$.