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Suppose that we have a free abelian group $F$. How can it be proved that $F$ has a subgroup of index $n$ which $n≥1$?

Honestly, according to the Theorems, I just know that if we take $X$ as a base for $F$, then $ F= \bigoplus_{\alpha \in X} \mathbb Z_\alpha \ $ in which for all $ \alpha \in X$; $\mathbb Z_\alpha \ $ is a copy of $ \mathbb Z $. What that subgroup could be? Thanks.

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    @M Turgeon: you meant $H$= $\bigoplus_{\alpha≠\alpha_{0}}{\mathbb Z_ {\alpha}}$$\oplus$$n\mathbb Z_{\alpha_{0}}$ ? Ok. I foundout what you all did.2012-05-16

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Theorem. Let $\mathfrak{V}$ be a variety of groups, and let $X$ be a nonempty set. The free $\mathfrak{V}$-group on $X$, $F_{\mathfrak{V}}(X)$ has a subgroup of index $n$ if and only if there is some $|X|$-generated group in $\mathfrak{V}$ with a subgroup of index $n$.

Proof. If $F_{\mathfrak{V}}(X)$ has a subgroup of index $n$, then it witnesses the existence of such a group. Conversely, let $G\in\mathfrak{V}$ be a group, with $\{g_x\}_{x\in X}$ a generating set of $G$, and suppose that $H$ is a subgroup of $G$ of index $n$. The map $f\colon X\to \{g_x\}_{x\in X}$ given by $f(x)=g_x$ induces, by the universal property, a surjective homomorphism $\mathfrak{f}\colon F_{\mathfrak{V}}(X)\to G$. By the isomorphism theorems, $H$ corresponds to a subgroup $\mathcal{H}$ of $F_{\mathfrak{V}}(X)$ that contains $\mathrm{ker}(\mathfrak{f})$, and hence $[F_{\mathfrak{V}}(X):\mathcal{H}] = [G:H] = n$, as claimed. $\Box$

Corollary. If $X\neq\varnothing$, then the free abelian group on $X$ has a subgroup of index $n$ for every positive integer $n$.

Proof. Let $C_n$ be the cyclic group of order $n$. This is $|X|$ generated, and contains a subgroup of index $n$ (namely, $\{1\}$). $\Box$

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    @Babak: There is no such thing as "**this** subgroup". There are *infinitely many* different subgroups of index $n$. Any subgroup of a free abelian group is necessarily free abelian, and a quotient of $F$ by a subgroup of index $n$ is **necessarily** of order $n$, and so in particular it cannot be torsion free (as it would be if it were $n\mathbb{Z}_{\alpha_0}$. You simply have completely misunderstood the comments above. It seems you just don't understand what is going on at all.2012-05-16