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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Structure of $(\mathbb{R}, +)$ as an abelian divisible group
It’s clear that $(\mathbb{R}, +)$ is a divisible group. By the structure theorem of abelian divisible groups, it follows that it’s isomorphic to a direct sum of copies of $\mathbb{Q}$, i.e., $\mathbb{...
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Structure of $p$-primary Abelian groups without Divisorial Elements
Let $A$ be a $p$-primary (in particular, torsion) Abelian group ($p$ prime). Assume that $A[p]=\{a \in A \vert pa=0\}$ is finite and that there are no nontrivial infinitely $p$-divisible elements, ie $...
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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 ...
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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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Commutative diagram involving divisible group
I am working on the exercise below from an old commutative algebra qualifying exam. I've been studying Atiyah & Macdonald's commutative algebra text.
Let $\pi:\mathbb{Q} \rightarrow \mathbb{Q}/\...
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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....
user775699
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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 ...
user775699
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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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Is Prüfer group divisible over the ring of $p$-adic integers?
Let $p$ be a prime number. Let $$\mathbb{Z}_{p}=\left\{\sum_{i=0}^\infty a_ip^i\mid a_i\in \{0,1,2,\dots,p-1\}\right\}~~and~~\mathbb{Z}_{p^\infty}=\left\{ \frac{a}{p^n}+\mathbb{Z}\mid a\in \mathbb{Z}\...
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The pure subgroups of a divisible abelian group are just the direct summands.
This is Exercise 4.3.3 of Robinson's "A Course in the Theory of Groups (Second Edition)". According to Approach0, it is new to MSE.
(NB: I have left out the modules tag for a reason: the ...
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Abelian divisible p-groups and Prüfer group
I've been asked to show tha if D is an abelian divisible p-group then is isomorphic to the sum of copies of $\mathbb{Z}(p^{\infty})$, which is to say that there exists a set $X$ such that $D\cong\sum_{...
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Largest divisible subgroup not an intersection
Is there an abelian group $G$ for which the largest divisible subgroup of $G$ (given by the sum of all divisible subgroups) is not the intersection of all subgroups of the form $nG$ over the positive ...
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Proving a Certain Subgroup of $\mathbb{Q}/\mathbb{Z}$ is Divisible
In an effort to compute the injective envelope of $\mathbb{Z}_p$ for $p$ prime, I need to show that the group $A \subset \mathbb{Q}/\mathbb{Z}$ generated by $\{1/p^r : r \in \mathbb{Z}^+\}$ is ...
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Prove that there is an $R$-module isomorphism between $Q \otimes_R N \cong N$, Q is a quotient field of $R$.
Let $R$ be an integral domain with quotient field Q and $N$ be a unitary, divisible, torsion-free left $R$-module. Show that there is an $R$-module isomorphism so that $Q \otimes_R N \cong N$.
Here ...
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A divisible abelian group is the direct sum of torsion subgroup and a torsion-free divisible subgroup.
Let $D$ be a divisible abelian group. It is known that $D \cong D_t \oplus D/D_t$, where $D_t$ is the torsion subgroup and $D/D_t$ is torsion-free and divisible. I want to find a torsion-free ...
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Divisible totally ordered additive abelian groups [closed]
Let $(G,+,\leq)$ be a divisible totally ordered additive abelian group and $g_{1},g_{2}\in G$. If for every integer $n>1$, $g_{1}\geq (1-\frac{1}{n})g_{2}$, Can we have $g_{1}\geq g_{2}$? Thanks ...
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An abelian, characteristically simple group is divisible (supposedly).
This is Exercise 4.13 of Roman's "Fundamentals of Group Theory: An Advanced Approach". According to this search and Approach0, it is new to MSE.
The Details:
On page 69 of Roman's book, we ...
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Find common number divisible by six different numbers
If there is recipe to find this - I would like to find the first common number divisible by the following six numbers- 260,
380, 460,560,760 and 960.
How does one calculate the numbers I need?
Any ...
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T.Y. Lam's divisible module definition and factor modules.
In surveying LMR of T.Y.Lam and get the divisible module (for any ring with unity not necessary an integral domain) definition as follows:
``A right $R$-module $I_R$ is called divisible if and only if ...
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Right exactness of quotienting out the maximal divisible subgroup
For every abelian groups $G$ let $\mathrm{d}G$ be its maximal divisible subgroup.
Then $G \mapsto G/\mathrm{d}G$ is a right exact functor $\mathbf{Ab} \to \mathbf{Ab}$.
Let
$$
0 \to G \...
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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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prove a field is a divisible group
I saw a statement: every field of characteristic 0, with its underlying additive group structure, is divisible.
I met with troubles in checking the above statement.
Suppose $\operatorname{char}(F)=...
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Showing groups are not divisible
Let $G$ be an abelian group and use additive notation. Call $G$ a divisible group if given x in G and a positive integer m we can always find an element y of $G$ such that $my = x$. For example, $\...
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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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Divisibility 1,2,3,4,5,6,7,8,9,&10
Tried:
Seems the ten-digit number ends with $240$ or $640$ or $840$ (Is not true, there are more ways the number could end)
$8325971640,$
$8365971240,$
$8317956240,$
$8291357640,$
$8325971640,$
$...
user608997
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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 ...
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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 ...
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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} \...
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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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If every nonzero quotient of $G$ is infinite, then $G$ is divisible
I tried proving this by contradiction (assuming $G$ is abelian, else $G$ could not possibly be divisible):
If $G$ is not divisible then there exist $g\in G$ and $n\in \mathbb {N}$ such that for all $...
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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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$(G,+)$ abelian group is divisible $\Longleftrightarrow$ it's an homomorphic image of $\Bbb Q^{(X)}$
Let $(G,+)$ be an additive abelian group.
Let us suppose $G$ divisible (i.e. $G=nG\;\;\;\forall n\ge1$). Let
then $x,y\in G$. Then there exists $z\in G$ and $n,m\ge1$ such that
$x=nz$ and $y=...
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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, ...
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$\mathbb{Z} [1/p] / \mathbb{Z}$ 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 $p$?
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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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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 ?
user123733