# Questions tagged [divisible-groups]

For questions about the structure and properties of a divisible group, which are Abelian groups in which one can "divide" by positive integers.

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### Abelian, divisible and finite subgroup implies trivial subgroup?

I'm reading a proof from my lecture and don't understand one step: We're given a group $(R,\cdot)$ and a proper subgroup $H$. We showed that $H$ must be finite $R$ is abelian and divisible The proof ...
• 29
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### Is possible a generalization of extending property of a divisible group?

I'm trying to generaliz this proposition: Let $G$ an abelian group and $D$ a divisible grop, for every $H\leq G$ and every homomorphism $f:H\to D$ exists $\overline{f}:G\to D$ an extension of $f$. I ...
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### Largest divisible subgroup of an abelian group

How do I prove that any abelian group $G$ contains divisible subgroup $H$, such that $G / H$ has no divisible subgroups other than $\{0\}$? Attempts: 1) Using Zorn's lemma was suggested to me in ...
• 656
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### Homomorphism from divisible group to finite group is always trivial

Let $A$ be a divisible group, let $B$ be a finite group, and let $f: A \rightarrow B$ be a homomorphism. Show that $f$ is trivial. (A group $A$ is divisible if for each $a \in A$ and $n \ge 1$ there ...
• 2,012
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### Show that $Q_p / \mathbb Z$ is divisible as $\mathbb Z$-module. [closed]

Let $p \in \mathbb N$ be a prime. Let Q_p : = \left \{ x \in \mathbb Q : (\exists k \in \mathbb Z)\ \mathrm {and}\ (\exists n \in \mathbb N)\ \mathrm {such}\ \mathrm {that}\ x= \frac {k} {p^n} \...
• 2,524
186 views

### Does every ordered divisible abelian group admit an expansion (and how many) to an ordered field?

Let $(G,+,<)$ be an ordered divisible abelian group. $1)$ Is it always the case that there exists a binary function $*:G\times G \rightarrow G$ such that $(G,+,*,<)$ is an ordered field? $2)$ ...
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• 11.9k
1 vote
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### Structure theorem for divisible modules

I would like to know if there is some analogue theorem of structure for divisible modules as there is for divisible abelian groups. More exactly, given a divisible $R$-module $M$, where $R$ is a PID, ...
• 6,167
1 vote
295 views

### $\mathbb{Z} [1/p] / \mathbb{Z}$ is divisible

I can't see why this group $\mathbb{Z} [1/p] / \mathbb{Z}$ is divisible. It is $p$-divisible, but how is it divisible by any integer $n$ when the denominator is only powers of $p$?
• 607
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### Unit group of an algebraically closed field is divisible

In the lecture notes on Valuation theory, in Ex. $1.16$ on page $11$ we are asked to show that: If $k$ is an algebraically closed field, then $k^{\times}$ is a divisible abelian group. Isn't \$k = \...
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