# 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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### Every torsion free divisible abelian group D is direct sum of the copies of the $\mathbb{Q}$

Prove that every torsion free divisible abelian group $D$ is direct sum of the copies of the $\mathbb{Q}$. If $a\in D$ then there exists unique $b \in D$ and $n\neq 0 \in \mathbb{Z}$ such that a=nb....
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### Properties of divisible groups

An abelian group $D$ is said to be divisible if given any $y \in D$ and $n\neq0 \in Z$, there exists $x \in D$ such that $nx = y$. (a) No nonzero finite abelian group is divisible. (b) No nonzero ...
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### An abelian torsion group has a unique basic subgroup iff it is divisible or bounded.

This is Exercise 4.3.14 of Robinson's "A Course in the Theory of Groups (Second Edition)". According to Approach0, it is new to MSE. The Details: Let $p$ be prime. A $p$-group is a group ...
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### Splitting of homomorphism onto Prüfer group

Let $G$ be a totally disconnected locally compact abelian group. Let $U \leq G$ be open and such that $G/U \cong \mathbb{Z}(p^\infty)$, the Prüfer $p$-group. In general $U$ is not necessarily a direct ...
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### Explanation on examples of non-divisible module

So the multiplicative group of complex numbers, $\mathbb{C}^*$, is divisible as an abelian group. Why is the multiplicative group of real numbers, $\mathbb{R}^*$, not divisible as an abelian group? ...
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### If $M$ and $N$ are divisible, abelian groups, show that their tensor product $M \otimes_\mathbb{Z} N$ is uniquely divisible

Recall that an abelian group $M$ is divisible if for each $m \in M$ and $r \in \mathbb{Z}$, there is an $m' \in M$ such that $rm' = m$. It is uniquely divisible if that $m'$ is unique. If $M$ and $N$ ...
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### $\mathbb{Z}_p$ has cohomological dimension 1

I would like to prove that $\mathbb{Z}_p$ has cohomological dimension 1, without using the fact that it is a free pro-p-group. If someone can suggest me any reference (or a hint for a proof) for ...
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### Dieudonné module associated to the dual of a $p$-divisible group

Let $k$ be a perfect field of characteristic $p>0$, and consider $X=(X_m,i_m)$ a $p$-divisible group of height $h$ over $\operatorname{Spec}(k)$: it is an inductive system where $X_m$ is a finite ...
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### Is $\mathbb{Z}_{p^\infty}$ a divisible group? [closed]

Let $$\mathbb{Z}_{p^\infty}=\{x\in \mathbb{Q}/\mathbb{Z}: \ \exists n\in \mathbb{N} ,\ \ p^nx=0\}.$$ Is $\mathbb{Z}_{p^\infty}$ a divisible group ?
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### On set of all $Z$-module homomorphisms as injective module

Let $\mathbb{Z}$ be the integers. The set $M$ of all $\mathbb{Z}$-module homomorphisms from a ring $R$ with unity to a divisible abelian group $A$ is known to be an injective left $R$-module. I am ...
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### Example of divisible direct product but not the direct sum

Here, divisible abelian group $G$ is defined such that for every $y\in G$ and nonzero integer $n$, we have $x\in G$ with $nx=y$. I am looking for an example where the direct product of groups is ...
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### Relationship between Archimedean and Divisible ordered groups

Let $(G,+,\leq)$ be a linearly ordered abelian group (i.e. the order is total and compatible with the sum) and $n\cdot x$ denote the classical action of $\mathbb{Z}$ over $G$ (i.e. $0$ for $n=0$, sum ...
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### Isomorphism between $\mathbb{T}$ and $\mathbb{R} \oplus \mathbb{Q}/\mathbb{Z}$.

I'm trying to prove that the circle group $\mathbb{T}$ is isomorphic to $\mathbb{R} \oplus \mathbb{Q}/\mathbb{Z}$ with a little bit of cardinal arithmetics. First, I know that $\mathbb{T}$ can be ...
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### Prove that$k^3n-kn^3$ is divisible by $6$ for all n∈N. [closed]

Hello I have problem with solution of task. Prove that $k^{3}n-kn^3$ is divisible by $6$ for all $n∈N$, $k∈N$ . Help me, please. I know, when $n^3-n$ is divisible by 6. $n^3-n= (n-1)(n)(n+1)$ and ...
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### Torsion-free, divisible multiplicative groups of fields

For which cardinalities $\kappa$ is $\def\Q{\mathbb Q}\Q^{(\kappa)}$—by which I mean the direct sum of $\kappa$-many copies of $\Q$—isomorphic (as an abelian group) to the multiplicative ...
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### Extension of divisible fields

Assume that $F$ is an infinite subfield of a field $K$ such that its multiplicative group, $F^\times$, is divisible. Also, $a\in K$ and $[F(a):F]<\infty$. Can we conclude that the multiplicative ...
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### Remainder when $2^{108}$ is divided by $11$? [closed]

What is the remainder obtained when $2^{108}$ is divided by $11$? I tried bringing in $11$ in the given no. such as in $(11-3)^{36}$ and then using binomial expansion...but its not helping. Any ...
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### If an abelian group $A$ is injective as $\mathbb{Z}$ module then $A$ is a divisible group

I was reading a proof to this proposition from this link: http://planetmath.org/abeliangroupisdivisibleifandonlyifitisaninjectiveobject they proceed by contradiction, so $A$ is not divisible and ...
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### Let G be a finite group. Let $a,b\in G$ be two distinct elements or order 2. Prove that if $ab=ba$, then the order of G is divisible by 4.

Let $G$ be a finite group. Let $a,b\in G$ be two distinct elements or order 2. Prove that if $ab=ba$, then the order of $G$ is divisible by 4.
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### Relation between open divisible subgroup and the quotient of the group with subgroup

I wanted to prove the following proposition: Let H be an open divisible subgroup of an abelian topological group G. Then G is topologically isomorphic to H x G/H. As for the proof, using extension of ...
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### A group is divisible if and only if it has no maximal subgroup ?

Is it true that a group is divisible if and only if it has no maximal subgroup ?
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### Irreducible subgroups of the additive rationals

Let $G$ be a group. A proper subgroup $H$ is called irreducible if $H$ can't be written as an intersection of two subgroups which contain it properly. I'd like to know if $(\mathbb Q,+)$ (and more ...
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### Divisible groups, exercise from Rotman's theory of groups

The following exercise is from Rotman, An Introduction to the theory of groups, 4th ed, p324. "The following conditions on a group G are equivalent: (i) G is divisible, (ii) Every nonzero quotient of ...
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### universal nonabelian divisible group

For this post, a group $G$ shall be referred to as generally divisible, in case $\forall{x\in G:}~\forall{n\in\mathbb{N}^{\times}:}~\exists{y\in G:}~y^{n}=x$. Note. Here is no commutativity ...
### What are the finite order elements of $\mathbb{Q}/\mathbb{Z}$?
I need to find what are the at the group $\mathbb{Q}/\mathbb{Z}$. I think that any element at this group has a finite order, but I don't know how to prove it... I'd like to get help with the proof ...
### Is $SO_n({\mathbb R})$ a divisible group?
The title says it all ... Formally, if $SO_n(\mathbb R)=\lbrace A\in M_n({\mathbb R}) |AA^{T}=I_n, {\sf det}(A)=1 \rbrace$ and $W\in SO_n(\mathbb R)$, is it true that for every integer $p$, there is ...