Subgroup containment relation in the group of integers equals divisibility relation on generators

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Statement

Let \mathbb{Z} denote the Group of integers (?) under addition. Suppose H,K are subgroups of \mathbb{Z}. Suppose H = m \mathbb{Z} and K = n \mathbb{Z}. Then, H \le K if and only if n| m, i.e., m is a multiple of n.

Note that, because every nontrivial subgroup of the group of integers is cyclic on its smallest element, the subgroups H and K can be written in the form m\mathbb{Z}, n\mathbb{Z} respectively.

Related facts

Corollaries

  • Given two subgroups m\mathbb{Z} and n\mathbb{Z}, their intersection is the subgroup generated by an element l with the property that m | l, n | l, and if m | k, n|k, then l|k. Such an integer l is termed a least common multiple of m and n (if we allow only nonnegative integers, then it is unique).
  • Given two subgroups m\mathbb{Z} and n\mathbb{Z}, their join is the subgroup generated by an element d with the property that d | m, d | n, and if c | m, c|n, then c | d. Such an integer d is termed a greatest common divisor of m and n (if we allow only nonnegative integers, then it is unique).
  • The greatest common divisor of m and n can be written as am + bn for some integers a and b. That is because it is in the subgroup generated by m\mathbb{Z} and n\mathbb{Z}.