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

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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