Theorem 1: Every subgroup of a cyclic group is cyclic.
Proof: Let be a cyclic group generated by . Let be a subgroup of . Now every element of , hence also of , has the form , with being an integer. Let be the smallest possible integer such that . We claim that . For this it is sufficient to show that , then for then . Now, if does not divide , then there exist integers and such that
Since , it follows that and hence its inverse .
But by supposition. Then from the above result it follows that , contrary to the choice of since was assumed to be the least positive integer such that . Therefore and so . But then
Thus every element of is of the form . Hence .
Theorem 2: Every subgroup of an infinite cyclic group is infinite.
Proof: Let be an infinite cyclic group. Let be a subgroup of . Then by the preceding theorem, where is the least positive integer such that . Now suppose, if possible, that is finite.
This implies that for some .
It follows that is of finite order and this in turn implies that is finite, contrary to the hypothesis. Hence must be an infinite cyclic subgroup of .