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.

112 questions
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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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 ...
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?