How to determine whether or not a group is a free group?

Question

How do i show whether or not a group could be free.

For example the Reals.

Answer 1

Here is a hint: when is $F(S)$, the free group on the set $S$, abelian? Now note that $\mathbb{R}$ is abelian...

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Suppose $\mathbb{R}$ is isomorphic to the free abelian group on a set $S$. Then any $a\in\mathbb{R}$ can be uniquely written as a finite sum $$a=\sum_{s\in S} n_ss,\quad n_s\in\mathbb{Z}$$ ("finite sum" meaning all but finitely many of the $n_s=0$).

Let $N=\max\{|n_s|\}$. We must also be able to uniquely write $\frac{a}{N+1}$ as a finite sum $$\frac{a}{N+1}=\sum_{s\in S}m_ss,\quad m_s\in\mathbb{Z}$$ But because $a=\sum_{s\in S}(m_s\cdot(N+1))s$, we must have that $n_s=m_s\cdot (N+1)$ for all $s\in S$ and $n_s,m_s\in\mathbb{Z}$, which is impossible (unless $a=0$) because $|n_s|<N+1$. Thus $\mathbb{R}$ cannot be a free abelian group.

Note: this argument works for any divisible group.