# 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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### Torsion in modules over unique factorisation domains [closed]

Let $R$ be a UFD and $R^n$ be the free $R$-module with basis $e_1,e_2,...,e_n$. Assume that the submodule $M_s$ is given by an $(n-1)\times n$ size matrix $A=[a_{i,j}]$, relations given by rows. Then ...
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### Torsion elements of $SL_3(\mathbb{F}_p[x])$? (Quick question)

Is every element of $SL_3(\mathbb{F}_p[x])$ a torsion element? Here are my thoughts: First of all, the group is noncommutative, so a torsion element is an element of finite order. I'm thinking of ...
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### For a PID $R$ prove that restriction-of-scalars induces equivalence of categories $\coprod_p (R_{(p)}-\mathrm{mod})_{tors}\to (R-\mathrm{mod})_{tors}$

Let $R$ be a PID. Prove that restriction-of-scalars induces an equivalence of categories of finitely generated modules $$\coprod_p (R_{(p)}-\mathrm{mod})_{tors}\to (R-\mathrm{mod})_{tors},$$ where the ...
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### Category of pair $(V,L)$ where $V$ is finite dim vector space and $L:V\to V$ is equivalent to the category of torsion finitely generated $K[T]$ module

Show that there is a canonical category of pairs $(V,L)$ consisting of a finite-dimensional $K$-vector space $V$ and a $K$-linear endomorphism $L : V \to V$, that is equivalent to the category of ...
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### Is $\mu_p$ the only nontrivial Galois module of size $p$? [closed]

Assume $p$ is prime. By $\mu_p$ I mean of course the $p$-th roots of unity. More specifically I'm working in the context of the $p$ torsion of elliptic curves. I guess having prime order narrows it ...
1 vote
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### Show that all the element of $\mathbb{Q} / \mathbb{Z}$ are of torsion.

I have a doubt conscerning the validity of my reasonning mostly the part "-2" . Question: We say that en element $g$ of a group $G$ is of torsion if it exists $n>0$ such that $g^n=e$ ($e$...
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### In what sense is cotorsion dual to torsion?

An $R$-module $M$ is called torsion if for every $m \in M$, there exists a non-zero divisor $r \in R$ s.t. $rm = 0$. If we let $S$ be the multiplicative system of non-zero divisors of $R$, we get that ...
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### Intersection between torsion cycles and free cycles

Let's consider a closed and oriented 4-manifold $M_4$ and denote $H_2(M_4,\mathbb{Z})$ as the homology group of 2-cycles and $Q_{M_4}(S_{A},S_{B})$ as the symmetric intersection pairing between 2-...
1 vote
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### Let $E:y^2=x(x-a)(x-b)(a,b\in \Bbb{Q})$ be an elliptic curve. What is $\#E_d(\Bbb{Q})_{tor}$?

Let $E:y^2=x(x-a)(x-b)(a\neq b\in \Bbb{Q})$ be an elliptic curve　over a field of rational numbers. Let $d\in \Bbb{Z}$ be a square free integer and $E_d:dy^2=x(x-a)(x-b)$ be a quadratic twist of $E$. ...
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### Decomposition of infinite abelian groups

A colleague recently mentioned a "Prüfer Decomposition Theorem", claiming that every abelian group $A$ could be expressed as the direct sum $A = T(A) \oplus F(A)$, where $T(A)$ is the ...
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### Maximal Essential Extension of Abelian Groups

Given any abelian group $A$, let $T(A)$ be its torsion subgroup and $F(A)=A/T(A)$. A homomorphism $\varphi:A\to B$ induces $T(\varphi):T(A)\to T(B)$, $F(\varphi):F(A)\to F(B)$, and $\varphi$ is one-to-...
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### $\prod_p \mathbb{Z}/p\mathbb{Z}$ is not the direct sum of $\bigoplus_p \mathbb{Z}/p\mathbb{Z}$ and a torsion-free subgroup

While I was reading "Abelian Groups" by Fuchs $(2015)$, I encountered Example $1.2$ in the chapter on Mixed Groups, which stated the following: Let $p_1,p_2,\dots,p_n,\dots$ denote different ...
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### $\underset{p\in \mathbb{P}}{\prod}\mathbb{Z}/p\mathbb{Z}$ is non splitting mixed abelian group.

We say that an abelian group $G$ is mixed if it has elements $\neq 0$, that are of finite order (torsion elements), as well as elements of infinite order (torsion-free elements). We denote the ...
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1 vote
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### Is the extension of a torsion free group by a torsion free group necessarily torsion free?

My instinct tells me the above isn't true, although I am having trouble finding a counterexample. Hints on where to look for a counterexample would be much appreciated. Hopefully there is some ...
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### Proof verification of decomposition of abelian group into torsion and free groups

Claim: Let $A$ be a (not necessarily f.g.) abelian group. Suppose that $T$ is the subgroup of all torsion elements in $A$, and we have that: $${A}\diagup{T} \cong F$$ where $F$ is a free abelian group....
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### Examples of an abelian group whose torsion subgroup is not a direct summand [duplicate]

According to Wikipedia there are abelian groups $G$ such that the short exact sequence $TG\to G\to G/TG$ is not split, where $TG$ is the torsion subgroup of $G$. However Wikipedia does not give any ...
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### Book on infinitely generated Abelian groups

I (believe to) understand "sufficiently well" finitely generated Abelian groups. As a consequence of looking on some direct limits of such groups, I am now however faced with infinitely ...
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### If $G$ is a torsion abelian group and $B_1, B_2$ are both basic in $G$, then $G/B_1$ is not necessarily isomorphic to $G/B_2$

Any two basic subgroups of $G$ are isomorphic ($B_1 \cong B_2$), but I am looking for a counterexample to show that the same is not true of the quotients $G/B_1$ and $G/B_2$. Here, the definition that ...
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1 vote
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### Question about Weil Pairing & the MOV attack

When I read any description of the Weil Pairing, it's described as a map between the additive groups $G1$ x $G2$ of an Elliptic Curve to a different group. $G1$ is the r-torsion group of the EC curve ...
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### How does the extension field $F_{p^k}$ contain complex roots of unity?

The Weil Pairing maps 2 points from the $m$-torsion subgroup of an Elliptic Curve over a Finite Field to an extension field of the Finite Field. Let $E$ be an elliptic curve over $F_p$ where $p$ is ...
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### Four torsion subgroups

I have started to solve some problems on Elliptic Curves, and I am stuck on a very simple exercise: https://math.mit.edu/classes/18.783/2022/ProblemSet1.pdf In "Problem 4. Four torsion subgroups (...
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### What is the proof of - $E[2]$ torsion group is isomophic to $Z_2 \oplus Z_2$?

This is from the book "Elliptic Curves" by Lawrence Washington. $E$ is an elliptic curve over $K$ $E[n] = \{P \in E(\bar K) \mid nP = \infty\}$ where $\bar K$ is the algebraic closure of $K$ ...
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### Proving that a torsion subgroup is normal

We prove, that the torsion group $T$, which has elements of finite order, $T\subset G$, is a normal subgroup of $G$. Definition: A subgroup T, is normal if $gTg^{-1}= T \ \forall g\in G$.\ UPDATE 2: ...
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### Structure of torsion subgroups in an elliptic curve with integer coefficients

Suppose we have an elliptic curve $E: y^{2} = (x+a)(x+b)(x+c)$ where $a,b,c \in \mathbb{Z}$. I want to know the structure of the torsion subgroup $E(\mathbb{Q})_{tors}$ based on the properties of the ...
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### Torsion point on elliptic curve over $\mathbb{Q},$ $E/\mathbb{Q}$ [duplicate]

Currently I am learning about elliptic curves. While playing with codes of elliptic curve in SageMath, I observed that if we take an elliptic curve $E: y^{2} = x^{3}+m,$ the torsion subgroup is ...
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### Torsion free groups with no unique products (notation)

I am reading a paper by William Carter titled "New examples of torsion-free non-unique product groups" and saw the following group: P_k=\langle a,b\mid ab^{2^k}a^{-1}b^{2^k},ba^{2}b^{-1}a^{...
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