# Free abelian group $F$ has a subgroup of index $n$?

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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2022-07-25 17:47:13
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