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# TIFR 2013 problem 5 | Non-Cyclic Subgroup of $\mathbb{R}$

Try this problem from TIFR 2013 problem 5 based on Non-Cyclic Subgroup of $\mathbb{R}$.

Question: TIFR 2013 problem 5

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.

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