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

From Groupprops

## Statement

Let denote the Group of integers (?) under addition. Suppose are subgroups of . Suppose and . Then, if and only if , i.e., is a multiple of .

Note that, because every nontrivial subgroup of the group of integers is cyclic on its smallest element, the subgroups and *can* be written in the form respectively.

## Related facts

### Corollaries

- Given two subgroups and , their intersection is the subgroup generated by an element with the property that , and if , then . Such an integer is termed a
*least common multiple*of and (if we allow only nonnegative integers, then it is unique). - Given two subgroups and , their join is the subgroup generated by an element with the property that , and if , then . Such an integer is termed a
*greatest common divisor*of and (if we allow only nonnegative integers, then it is unique). - The greatest common divisor of and can be written as for some integers and . That is because it is in the subgroup generated by and .