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

For example the Reals.

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1 Answer 1

up vote 8 down vote accepted

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


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.

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2  
Maybe @Zarks meant free abelian? –  t.b. May 8 '11 at 10:45

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