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Questions tagged [torsion-groups]

For questions about torsion groups and their properties. A torsion group is a group in which every element has finite order.

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isogenous elliptic curves have same rank

This is based on exercise 14.3 from Cassels, Lectures on Elliptic Curves. Let $$E:y^2=x(x^2+ax+b), E':y^2=x(x^2+a_1x+b_1)$$ be two elliptic curves over $\mathbb{Q}$, with $a_1=-2a$, $b_1=a^2-4b$. We ...
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27 views

Infinite finitely generated amenable periodic groups

I know that the Grigorchuk group is an example of this. I also know that there are other Grigorchuk groups that satisfy this as well. Are there any other examples? Is any general structure/...
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49 views

Non-abelian group, where torsion elements form subgroup

We proved in the lectures, that for an abelian group, the torsion elements (elements of finite order) form a subgroup. I also found an example for a non abelian group, where the torsion elements do ...
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45 views

Elliptic Curve Division Points

There is a statement about the number of division points, which I've read in a few papers, but it never seems to have any references where it comes from or why it is true. The statement is the ...
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1answer
28 views

Enough injectives in the category of torsion abelian groups

Claim: The category of torsion abelian groups has enough injectives. I thought I had a proof of this, but discovered a mistake in my proof. I was trying to use the techniques of the usual proof that ...
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1answer
95 views

$\operatorname{Frac}(A)/A$ as an $A$-module

I am wondering about a question: We know that $\mathbb{Q}/\mathbb{Z}$ is torsion group and $\mathbb{Q}/\mathbb{Z}=\bigoplus_{p\text{ prime}}\mathbb{Q}/\mathbb{Z}(p)$ where $\mathbb{Q}/\mathbb{Z}(p)=\{...
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45 views

The n-torsion subgroup $E[n]$ is isomorphic to $Z_n$ x $Z_n$

In Washington's "Elliptic Curves: Number Theory and Cryptography", the proof that $E[n] \simeq Z_n$ x $Z_n$ is concluded by referring to the structure theorem of finite abelian groups ($E[n] \simeq ...
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Show that if $y$ has finite order, then $x^ny=yx^n$ for some $n\in\Bbb{N}^{\ast}$ and all $x$.

Let $G$ be a group and $F$ the set of finite order elements in $G$. If $F$ is finite, prove that there exists $n\in \mathbb{N}^{\ast}$ such that $x^ny=yx^n$ for all $x\in G,y\in F$. My progress so ...
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35 views

When is $\mathbb{Z}/m\mathbb{Z}$ a module over $\mathbb{Z}/n\mathbb{Z}?$

I'm trying to find a torsion free finite module: let $M$ be such $R$-module. If for every $0\ne a\in M$ and every $0\ne r\in R$ we have $ra\ne 0$ then $$ra=sa\implies (r-s)a=0\implies r=s$$ so the ...
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If $G=A\times B$, where $A$- torsion group, $B$-free abelian group, how to show that $A=G_{tor}$

$G$ is a finitely generated Abelian group with $G=A\times B$ where $A$ is a torsion group and $B$ a free Abelian group. I know that every finitely generated Abelian group can be factored into cyclic ...
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36 views

Torsion in Cokernel is Annihilated by Integer

My question refers to an argument in a proof from T. tom Dieck's "Algebraic Topology and Transformation Groups" (Lemma 3; page 2). Here the excerpt and the red tagged step which I don't understand: ...
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58 views

Generating random (torsion) point on elliptic curve efficiently

I am looking for an efficient way to generate a random point on an elliptic curve over a finite field, $E(\mathbb{K})$. I know that you can pick a random $x$, compute e.g. in Weierstrass coordinates ...
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55 views

Question in Algebra by Serge Lang.

The following is a lemma in Algebra (page 44): $A$ is a finite abelian p-group. Let $\overline{b}$ be an element of $A/A_1$ ($A_1$ is a cyclic group generated by $a_1 \in A$ of period $p^{r_1}$), of ...
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2answers
48 views

Let $A$ be an abelian finitely generated free group and $A/B$ be a torsion group. Show that $rank(A)=rank(B)$.

Let $A$ be an abelian free group that is finitely generated, and let $B\subset A$ be a subgroup of $A$ such that $A/B$ is a torsion group. Show that $rank(A)=rank(B)$. From the hypothesis, I know ...
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91 views

Isomorphism on a torsion group - automorphism or endomorphism?

Let $f:G\to G$ be a surjection from a torsion group $G$ onto itself. Let the kernel have infinite cardinality: $\lvert\ker(f)\rvert=\aleph_0$ What category of function on groups is this? To my ...
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83 views

Elliptic curve with only torsion points

Is it possible to have an elliptic curve $E$ over $\Bbb Q$ such that $E( \overline{\Bbb Q})$ is a torsion abelian group? I know that $E(\Bbb Q)$ can be a finite group. I know that $E(\Bbb C)$ is a ...
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24 views

How do I interpret *all factors have a common finite exponent* in this context?

"The product of infinitely many torsion groups will no longer be a torsion group unless all factors have a common finite exponent (which is not the case if we take Prüfer groups)." How do I interpret ...
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25 views

Divergence of a third rank tensor with non symmetric connection

I need to construct a 2 rank tensor with covariant derivative of a 3 rank tensor. Like that: $${F^{\mu}}_{\nu} = \frac{1}{2} \nabla_{\rho} {{J^{\mu}}_{\nu}}^{\rho}$$ Thus $${F^{\mu}}_{\nu} = \frac{...
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136 views

Torsion subgroup of a finitely-generated abelian group is finite?

The above claim was made at the very beginning of a proof of the structure theorem for finitely-generated abelian groups and brushed off as easy. However, I think the problem is easy if the torsion ...
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115 views

What are the elements of $\Bbb Z[\frac16]/{\sim}$ and what do the subgroups and orders of elements look like?

What are the elements of $\Bbb Z[\frac16]/{\sim}$ and what do the subgroups and orders of elements look like? In which $\exists i\in \Bbb Z:4^ia=b\implies a\sim b$ What's the identity element, for ...
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Can we prove that $(\mathbb{Z}/p^{n}\mathbb{Z})^{\times}$ is a cyclic group by using $p$-adic integer?

It is well known that $(\mathbb{Z}/p^{n}\mathbb{Z})^{\times}$ is a cyclic group for a prime $p>2$ and $n\geq 1$. However, most of the proofs are a little complicated, and I want to find some neat ...
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2answers
53 views

Proving Specific Quotient Group is Torsion-Free

I am trying to prove that the quotient group $(\mathbb{Z} \oplus \mathbb{Z})/(\mathbb{Z} \cdot (11,13))$ is torsion-free. I know how I have to show that the only element in the group that has finite ...
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1answer
114 views

Torsion Coefficient in Group Theory

I have seen a calculation about torsion coefficient Determining torsion coefficients But I am now facing another similar question and not sure about is my answer correct. The question: Find the ...
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1answer
56 views

About infinite chains of subgroups containing an infinite subgroup of a locally finite group

Let $G$ be an infinite locally finite (non-solvable) group, let $\{H_i\}_{i\in\mathbb{N}}$ be a strictly totally ordered family of subgroups of $G$ and let $H$ be the intersection of that family. If $...
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53 views

When do we know a quotient of the polynomial ring over a local ring is torsion free

Let $R$ be a local ring (e.g. the discrete valuation ring $\mathbb C[[T]]$) and $\mathfrak{m}$ its maximal ideal. Consider the polynomial ring $R[X_1,\dots, X_n]$ in $n$ variables and a finitely ...
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1answer
116 views

$2$-torsion subgroup of $M$ v.s. $M/(2M)$

From this question/answer, https://math.stackexchange.com/a/2843091/141334, I learned the word that $T$ denotes the $2$-torsion subgroup, i.e., the subgroup comprising elements of order dividing $2.$ ...
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1answer
72 views

Can a countably infinite torsion group $G$ exist in which $G=F\times\{0,1,2\}$ and $G^2=F\times\{1,2\}$?

Can a countably infinite torsion group $G$ exist in which $G=F\times\{0,1,2\}$ and $G^2=F\times\{1,2\}$? OR (and perhaps this is a better description, but I don't quite know enough about countably ...
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Can a countably infinite torsion group exist in which the density of squares is double the density of non-squares?

Can a countably infinite torsion group exist in which the density of squares is double the density of non squares? By $x\in G$ being a square I simply mean there exists $y\in G$ satisfying $y\cdot y=...
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1answer
79 views

Algebraic closure for torsion points on elliptic curves

In my book about elliptic curves I've read about torsion points of an elliptic curve $E$ defined over $K$, for $n$ positive: $E[n]=\{P\in E(\overline{K})\mid nP=\infty\}$ I've got two questions ...
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1answer
43 views

On finitely generated torsion $R$-modules $M$ when $R$ is commutative.

Let $R$ be a commutative ring and let $M$ be a finitely generated torsion $R$-module. If for any $x\in M$, we have that $R/\operatorname{ann}(x)$ is simple, I want to prove that $$M\cong \bigoplus_{i=...
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Computing the Torsion Subgroup of $E(\mathbb{Q})$, where $E:Y^2=X^3 +2$

I have to compute the Torsion Subgroup of $E(\mathbb{Q})$, where $E:Y^2=X^3 +2$ using Nagell-Lutz. So, if $(x,y) \in E(\mathbb{Q})_{\text{tors}}$ then $x,y \in \mathbb{Z}$ and either $y=0$ or $y^2 \ \...
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1answer
163 views

Direct limit of torsion free abelian groups is torsion free.

I am trying to show that if $G_{\alpha}$ form a directed set of torsionfree abelian groups, then $\varinjlim G_{\alpha}$ is torsionfree abelian. One approach I thought of is to use a Zorn's lemma ...
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65 views

Are these two functors, full functors?

Ler $\mathcal{G}$, $\mathcal{G_0}$ and $\mathcal{G_{\infty}}$ be the category of abelian groyps, torsion abelian groups and infinite abelian groups, respectively. The morphism of the first and second ...
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74 views

The shape and group structure of an elliptic curve over $\overline{\mathbf{F}_p}$ and intermediary extensions

Let $p$ be prime, let $q$ be a power of $p$ and let $E/\mathbf{F}_q$ be an elliptic curve defined over the finite field $\mathbf{F}_q$. Let $\overline{\mathbf{F}_q}$ be the algebraic closure of $\...
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1answer
160 views

Classification of finite rank Abelian groups

An Abelian group $A$ is said to be finite rank if there is a natural number $n$ such that any finitely generated subgroup of $A$ can be generated by no more than $n$ elements. It is well-known that a ...
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Torsion subgroup of $\mathbb{Z}\times \left( \mathbb{Z}/n\mathbb{Z}\right)$

An exercise from Dummit and Foote, example 2.1.7 Fix some $n\in \mathbb{Z}$ with $n>1$. Find the torsion subgroup of $\mathbb{Z}\times \left( \mathbb{Z}/n\mathbb{Z}\right)$. Show that the set of ...
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1answer
46 views

Functions used in the Weil pairing

I'm using the definition of the Weil Pairing from Silverman's book on elliptic curves. When he defines the functions used in the pairing, he writes: I have two related questions about what is written ...
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1answer
40 views

A finite group acting on a torsion-free group

I'm trying the following problem Let $G$ be a group. $H, K$ subgroups of $G$, where $|K|=m$, $H$ is torsion-free and $[G:H]=n$. Prove that $m\leq n$ and $m\mid n$. The hint: Use some group action ...
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Examples of torsion trace and torsion reject

I am trying to understand the following examples in more details. Why are the following hold?? Can we prove it in details? Let the class $$Y = \{Z_n:n=2,3,\cdots\},$$ then for each abelion group, ...
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Weil pairing on $E[p]$ is trivial

I'm currently working through Elliptic Curves, L. C. Washington. On page 147 he writes "The Weil pairing is not defined on $E[p]$ (or, if we defined it, it would be trivial since $E[p]$ is cyclic and ...
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Why If $f: A \rightarrow B$ is an $R$- module homomorphism then $f(T(A)) \subset T(B)$?

Why If $f: A \rightarrow B$ is an $R$- module homomorphism then $f(T(A)) \subset T(B)$?, where $T(A)$ is called the torsion submodule. Could anyone clarify this for me please?
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How to compute largest divisor in exterior product

I could use help on the following problem: Suppose $v_1, \dots, v_k$ are linearly independent in $\mathbf{Z}^n$. Then show that the cardinality of the torsion subgroup of $\mathbf{Z}^n/\langle v_1, \...
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1answer
60 views

Points with 0 $x$-coordinate in $r$-torsion and distortion map

Suppose I have the curve $E/F_{59}: y^2 = x^3+1$ -i.e. it is supersingular. Hence number of points is $\#E(F_q) = 59+1$ and for $r=3$ (i.e $3\ |\ 60$ but $3^2\not|\ 60$), the embeding degree is $k=2$ ...
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155 views

Trying to construct an elliptic curve pairing (Tate) example. Can someone help me figure out where my mistake is?

I'm trying to construct an example of the Tate pairing, but for some reason I'm not getting the bilinearity property that I should be getting. Here's my setup: Take $q=29$ and $$E/\mathbb{F}_q : y^2 =...
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122 views

Difference between torsion and Lie bracket

After viewing a lecture on torsion, the lecturer said that the torsion is the failure of curves to close. Since this is almost also what I have read about the Lie bracket, I want to know their ...
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1answer
227 views

Why is the group cohomology for a profinite group always torsion?

Let $G$ be a profinite group, $A$ be a discrete $G$-module, and $n>0$ be an integer. Why is the cohomology group $H^n(G;A)$ a torsion abelian group? Here $H^n$ denotes the continuous cohomology ...
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64 views

Torsion points of elliptic curves

I'm studying "The Arithmetic of Elliptic Curves" by Silverman and I'm having trouble with the torsion subgroup. Corollary 6.4 at page 86 states that the subgroup $E[m]$ which is the $m$-torsion ...
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1answer
148 views

$\operatorname{Tor}_1^\mathbb{Z}(U,U)=0$. Is $U$ torsion free?

Let $U$ be a $\mathbb{Z}$-module with $\operatorname{Tor}_1^\mathbb{Z}(U,U)=0$. Does this imply that $U$ is torsion free? I know that if $\operatorname{Tor}_1^\mathbb{Z}(U,M)=0$ for every $\mathbb{Z}$...
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34 views

Sage: Torsion parameter in elliptic_curves db

I'm cross-posting from StackOverflow because I didn't get any answers Looking at the Tables of elliptic curves of a given rank documentation page and at Mazur's Theorem, a question pops out at me: ...
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105 views

trivial group torsion or torsion-free?

Forgive me for this question: Is the trivial group $\{id\}$ both, torsion free and a torsion group? Or how is the convention here? By definition (https://en.wikipedia.org/wiki/Torsion-...