Every nontrivial discrete subgroup of reals is infinite cyclic

Statement

Let $H$ be a nontrivial discrete subgroup of $\mathbb {R}$, the additive group of real numbers. Then, $H$ is an infinite cyclic group; it is described as $m\mathbb {Z}$ where $m$ is the smallest positive real number in $H$.

Related facts

• Every nontrivial subgroup of the group of integers is cyclic on its smallest element
• Every discrete subgroup of Euclidean space is free Abelian on a linearly independent set

Facts used

• Discrete subgroup implies closed: Any discrete subgroup of a $T_{0}$-topological group is closed.

Proof

Given: A nontrivial discrete subgroup $H$ of $\mathbb {R}$

To prove: $H$ is an infinite cyclic group, generated by $m$, where $m$ is the smallest positive real number in $H$

Proof: Since $H$ is nontrivial, it contains a nonzero element. Further, since $H$ is a subgroup, it contains a nonzero positive element (if the element we start with is negative, we can take its additive inverse).

Consider the set of all positive elements of $H$. This is a nonempty set, so it has an infimum. Moreover, since $H$ is discrete, there exists a neighbourhood of $0$ not intersecting $H$, hence the infimum is strictly greater than $0$. Since $H$ is a closed subgroup of $\mathbb {R}$, this infimum must itself be in $H$, and is hence the smallest element of $H$. Call this element $m$.

Now suppose $a$ is any element of $H$. We can write:

$a=mq+r$

where $q$ is an integer and $0\leq r<m$. Then, $r=a-mq\in H$, since $a,m\in H$. Since $m$ is the smallest positive real in $H$, $r=0$. Thus, every element of $H$ is a multiple of $m$, completing the proof.