Every nontrivial subgroup of the group of integers is cyclic on its smallest element

Statement

Let $H$ be a subgroup of $\mathbb{Z}$, the group of integers under addition. Then, there are two possibilities:

• $H$ is the trivial subgroup, i.e. $H=\{0\}$
• $H$ contains a smallest positive element, say $m$, and $H$ is the set of multiples of $m$. Thus, $H$ is an infinite cyclic group generated by $m$, and is isomorphic to $\mathbb{Z}$. We typically write $H=m\mathbb{Z}$.

Related facts

The result actually generalizes to the additive group of any Euclidean domain, where smallest element is replaced by element of smallest norm. A difference in the general case is that we may not have a way of uniquely picking one generator for the subgroup.

Another generalization is to discrete subgroups of the reals:

Every nontrivial discrete subgroup of reals is infinite cyclic

Proof

Given: A nontrivial subgroup $H$ of $\mathbb{Z}$, the group of integers under addition

To prove: There exists a smallest positive element $m$ in $H$, and $H=m\mathbb{Z}$, so $H$ is isomorphic to $\mathbb{Z}$

Proof: First, observe that if $H$ is nontrivial, then there exists a nonzero element in $H$. This element may be either positive or negative. However, since $H$ is a subgroup, it is closed under taking additive inverses, so even if the element originally picked was negative, we have found a positive number in $H$.

Thus, the set of positive numbers in $H$ is nonempty. Hence, there exists a smallest positive number in $H$. Call it $m$.

Clearly, all the integer multiples of $m$ are in $H$. We need to prove that every element in $H$ is a multiple of $m$.

By the Euclidean division algorithm, we can write:

$n=mq+r$

where $q,r$ are integers and $0\le r<m$. Since $n,q\in H$, $r=n-mq=n-(q+q+\dots +q)\in H$. Thus, $r$ is a nonnegative integer less than $m$ such that $r\in H$. By the minimality of $m$, we have $r=0$, so $m|n$, as desired.

Thus, $H=m\mathbb{Z}$, or $H$ is the set of multiples of $m$.

An explicit isomorphism from $\mathbb{Z}$ to $H$ is given by the map sending an integer $x$ to the integer $mx$.

References

Textbook references

• Algebra by Michael Artin, ISBN 0130047635, 13-digit ISBN 978-0130047632, ^{More info}, Page 45, Proposition 2.4, Chapter 2, Section 2