# Questions tagged [free-abelian-group]

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### Set of homomorphisms on a free abelian group is a free abelian group.

If $G$ is a free abelian group with rank $n$, I need to show that ${\rm Hom}(G,\mathbb{Z})$, set of all homomorphisms is also free abelian group of rank $n$, My work: Since $G$ is free abelian group ...
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### Is every surjective group homomorphism $f:\mathbb{Z} \oplus \mathbb{Z} \rightarrow \mathbb{Z} \oplus \mathbb{Z}$ also injective? [duplicate]

I would like to know if every surjective group homomorphism $f:\mathbb{Z} \oplus \mathbb{Z} \rightarrow \mathbb{Z} \oplus \mathbb{Z}$ is also injective. I suspect it is true, but I'm not sure how to ...
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### Is $\mathbb{Z_5}$a free abelian group ? Yes/No [closed]

Is $\mathbb{Z_5}$ a free abelian group ? My attempt: I think $\mathbb{Z_5}$ is free abelian group By the definition of free abelian group $X$ generates $G$, and $n_1x_1 +n_2x_2 +\dots+n_rx_r=0$ ...
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### Example/Demonstration of Quotient group $G/ H$ of Free Abelian Group
Let $G$ be a free abelian group of rank $r$, and $H$ a subgroup of $G$. Then $G/ H$ is finite if and only if the ranks of $G$ and $H$ are equal. If this is the case, and if $G$ and $H$ have $\Bbb Z$-...
Every subgroup $H$ of a free abelian group $G$ of rank $n$ is free of rank $s \leq n$. Moreover there exists a basis $u_1, ... , u_n$ for $G$ and positive integers $\alpha_1, ... ,\alpha_s$ such that \$...