# Questions tagged [free-abelian-group]

This is for questions about abelian groups, each with a basis.

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### Direct sum of free abelian group and quotient of abelian group by subgroup

I'm currently studying abelian groups in Kurosh's The Theory of Groups. I'm trying to understand the proof of the theorem: Let $B \leqq A$ be abelian groups. If $A/B \cong C$ and $C$ is a free group, ...
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### Proof that any element of a free abelian group which is not divisible by any $k>1$ can be extended to a basis

Let $A$ be a free abelian group of rank $n$, and let $\alpha_1 \in A\setminus \{0\}$ such that $\alpha_1 \not \in kA$ for all $k > 1$. Do there always exist $\alpha_2,\ldots, \alpha_n \in A$ such ...
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### Order of a quotient of a free abelian group

Let $G\subseteq \mathbb{C}$ be a free abelian group of rank $n$ and let $p$ be a prime. Then, we know that $$|G/pG|=p^n$$ and in fact for this we don't even need $p$ to be a prime. Suppose now that ...
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### Semidirect product of free abelian groups

Consider the semidirect product $$G=\mathbb{Z}^n \rtimes \mathbb{Z}^m$$ Is $G$ always virtually abelian? Is it the case that the abelianization of $G$ is $\mathbb{Z}^{n+m}$ if and only if $G$...
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