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

This article adapts material from the main article: every nontrivial subgroup of the group of integers is cyclic on its smallest element

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WHAT YOU NEED TO DO:

• Read the statement and understand it thoroughly.
• Go through the proof very carefully. Make sure you understand all the steps of the proof, particularly the application of Euclidean division.

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 $H = m\mathbb{Z}$.

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 $mx$.

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