Non-Cyclic Subgroup of $$\mathbb{R}$$ (TIFR 2013 problem 5)

Question:

True/False?

All non-trivial proper subgroups of $$(\mathbb{R},+)$$ are cyclic.

Hint: What subgroups comes to our mind immediately?

Discussion: $$(\mathbb{Q},+)$$ is a subgroup of $$(\mathbb{R},+)$$. Is $$(\mathbb{Q},+)$$ a cyclic group?

Suppose $$(\mathbb{Q},+)$$ is cyclic. Then there exists a generator say $$\frac{a}{b}$$. Note that, we are only allowed to use addition (and subtraction) to create $$\mathbb{Q})\ Therefore, we can create  \frac{a}{b}+\frac{a}{b}+…+\frac{a}{b}=n\frac{a}{b}=\frac{na}{b}  Also, we can create  (-\frac{a}{b})+(-\frac{a}{b})+…+(-\frac{a}{b})=n(-\frac{a}{b})=-\frac{na}{b}  Notice that we can increase the magnitude of the numerator, but not the magnitude of the denominator. For example, we cannot create \(\frac{a}{2b}$$ using $$\frac{a}{b}$$ and the binary operation +.

Therefore, $$(\mathbb{Q},+)$$ is not cyclic.

Remark: There is one result which states that subgroups of $$(\mathbb{R},+)$$ are either cyclic or dense. Notice that although $$(\mathbb{Q},+)$$ is not cyclic it is dense.