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 ...
Jupiter's user avatar
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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 ...
June in Juneau's user avatar
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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 ...
Squirrel-Power's user avatar
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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 ...
Squirrel-Power's user avatar
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Suppose $\mathbb{Q}(E[N]) = \mathbb{Q}(\zeta_N)$ and let $P$ be a rational point of order $N$, show $E[N]\cong \mathbb{Z}/N\mathbb{Z}\times \mu_N$

$N$ is prime and $E$ is of course an elliptic curve defined over $\mathbb {Q}$. My attempt so far: Consider the Galois action of $G = Gal(\mathbb{Q}(\zeta_N)/\mathbb{Q})\cong (\mathbb{Z}/N\mathbb{Z})^\...
Camilo Gallardo's user avatar
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Size of torsion subgroup of an abelian variety

Fisrt, let $k$ be a number field, and let $A/k$ be an abelian variety. I know that by the Mordell–Weil theorem, $A(k)$ is finitely generated, but I saw here (in the introduction) that $A(k)_{tors}$ is ...
Or Shahar's user avatar
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Why is $\wedge ^2 E[p] \cong \mu_p$?

As the title says, why is $$\wedge ^2 E[p] \cong \mu_p,$$ where $E[p]$ refers to the $p$-torsion points of an Elliptic curve over a number field $K$, $\wedge$ refers to the exterior product and $\...
Yang Awotwi's user avatar
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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 ...
Camilo Gallardo's user avatar
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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 $g^n=e$ ($e$...
OffHakhol's user avatar
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Determining the size of an $m$-torsion group of elliptic curves over finite fields

In this paper by De Feo the following is stated (in proposition 4): Let $E$ be an elliptic curve defined over a field $k$, and let $m\neq 0$ be an integer. The $m$-torsion group of $E$, denoted by $E[...
jorisperrenet's user avatar
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Torsion spin-c structures on $S^1\times S^2$

I have been reading the paper https://arxiv.org/pdf/1902.04050.pdf by Zemke and at some point we have the following: My question is, how can one make sense of torsion $\mathrm{Spin}^c$ structures on $...
horned-sphere's user avatar
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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-...
JQ Skywalker's user avatar
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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 ...
John123's user avatar
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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 ...
John123's user avatar
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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 ...
user1044791's user avatar
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1 answer
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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 ...
Carla_'s user avatar
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relationship between Torsion and Relations [closed]

We know that every abelian group $H$ has a presentation of the form $\langle S |R\rangle$, where $S$ are the generators and $R$ are the relations. Intuitively there should be some connection between $...
Csaba Daniel Farkaš's user avatar
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Does there exist a torsion-free compact Hausdorff topological group?

Let $G$ be a compact Hausdorff topological group. Does $G$ have torsion elements? I couldn't find examples of torsion-free compact Hausdorff topological groups. If $G$ is a Lie Group it should be true....
psl2Z's user avatar
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Quotient of an abelian group by torsion subgroup provides a characteristic free abelian subgroup? [closed]

I have a follow up to this question: Rank of the quotient of an Abelian group by its torsion part?. So my understanding is given a finitely generated abelian group $G$ and its torsion subgroup $T_G$ ...
Txim's user avatar
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1 answer
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Can't understand matrix representation of a torsion subgroup homomorphism

In the book Elliptic curves number theory and cryptography (Lawrence C. Washington 2nd eddition), page 79, there is a theorem (Theorem 3.2): Let $E$ be an elliptic curve over a field $K$ and let $n$ ...
mildog8's user avatar
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Is $G=\{ e\}$ both torsion group and a torsion-free group?

Is $G=\{ e\}$ a torsion group or a torsion-free group? Because all elements in $G$ has finite order, it is a torsion group. Because $e$ is the only element in $G$ with finite order, it is a torsion-...
MathFail's user avatar
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Number of given order subgroups of the torsion subgroup (Elliptic Curve)

I'm studing the paper "ON THE COST OF COMPUTING ISOGENIES BETWEEN SUPERSINGULAR ELLIPTIC CURVES" (link) and, at some point, autors say that (assuming $e$ even): the number of order-$\ell^{e/...
Manuel Bravi's user avatar
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0 answers
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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 ...
NeitherNor's user avatar
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3 votes
1 answer
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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 ...
I Eat Groups's user avatar
1 vote
1 answer
108 views

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 ...
user93353's user avatar
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3 votes
1 answer
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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 ...
user93353's user avatar
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1 vote
2 answers
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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 (...
Tireless and hardworking's user avatar
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1 answer
40 views

notation $tA$ in algebraic topology

Currently, I am studying torsion group. Reading the lecture notes, I found this theorem Let $R$ be an integral domain and $Q:= \operatorname{Frac}(R)$ with $K=Q/R$, then $\operatorname{Tor}_1^R(K,A) \...
phy_math's user avatar
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5 votes
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The quotient of totally ordered additive abelian groups which is torsion (value groups in valuation theory) [closed]

Let $(G,+,\leq)$ be a totally ordered additive abelian group and $(H,+\leq)$ be a subgroup of $(G,+,\leq)$. Let the quotient $\frac{G}{H}$ be torsion and be of finite order $n$ ($|\frac{G}{H}|=n$), ...
Mary's user avatar
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$\phi$-torsion of a group

Let $\phi:G\to G$ be an endomorphism of an abelian group $G$. If $\phi$ is defined by $g\mapsto g^n$, then I already know that $G[\phi]:=G[n]=$ $n$-torsion of $G=\ker(\phi)$. But if $\phi$ is a ...
Or Shahar's user avatar
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Discrepancy between order of the identity element & its inclusion in n-torsion sets

The order of the identity element of a group is usually considered to be one because $1 * e = e$ or $e^1 = e$. However, the text on torsion points (from Mathematical Cryptography by Silverman, Pipher ...
user93353's user avatar
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2 votes
2 answers
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Showing torsion abelian groups have a unique $\widehat{\mathbb{Z}}$-module structure.

Let $E$ be a torsion abelian group, we write it additively. Define the multiplication \begin{align} \widehat{\mathbb{Z}} &\times E \rightarrow E, \\ (a,g) &\mapsto a\cdot g:=a_ng, \end{align} ...
carraig's user avatar
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1 vote
0 answers
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Prove 3-torsion in characteristic 2 field is isomorphic to $\mathbb{Z}_3 \times \mathbb{Z}_3$ [duplicate]

From washington's book exercise 3.2. I have to show that $E[3] = \mathbb{Z}_3 \times \mathbb{Z}_3$ where $E$ is defined over a characteristic 2 field. Given the generalized Weierstrauss equation $y^2 +...
Ignatio Mobius's user avatar
2 votes
2 answers
101 views

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 $K$ ...
user93353's user avatar
  • 466
1 vote
1 answer
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Injection of rational torsion points over elliptic curve over integers in to $\Bbb F_p$-torsion points via finite flat group schemes

Suppose $X$ is an elliptic curve over $\Bbb Q$ defined by a Weierstrass equation. We may consider this with integer coefficients and then reduce modulo $p$, and if $p$ is nice enough with respect to $...
Hank Scorpio's user avatar
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4 votes
1 answer
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Let $A$ be abelian of order $m^2$ with $|A[d]|=d^2$ for every $d\mid m$. Prove $A\cong \Bbb{Z}/m\Bbb{Z}\times\Bbb{Z}/m\Bbb{Z}$

While studying elliptic curves I was trying the following problem related with group theory: Let $A$ be an abelian group of order $m^2$ with $|A[d]|=d^2$ for every $d\mid m$. Prove that $A\cong \...
Marcos's user avatar
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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: ...
Superunknown's user avatar
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1 vote
1 answer
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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 ...
Atratrana Suna's user avatar
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1 answer
157 views

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 ...
Atratrana Suna's user avatar
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1 answer
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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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1 answer
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Elliptic curve over $\mathbb{Q}$ with two distinct non-torsion points $P$ and $Q$ such that $nP \neq mQ$ for all $n,m \in \mathbb{Z}-\{0\}$. [closed]

We know an elliptic curve over $\mathbb{Q}$ can have at most finite number of torsion points. The torsion group for $E$, $E_{tors}$ is either $\mathbb{Z}/n\mathbb{Z}$ with $1 \leq n \leq 10$ or $n = ...
Atratrana Suna's user avatar
1 vote
1 answer
157 views

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 ...
Shaun's user avatar
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If $G$ is abelian and ${\rm Aut}(G)$ is finite, prove that $G$ has finite torsion subgroup.

This is Exercise 4.3.10 of Robinson's "A Course in the Theory of Groups (Second Edition)". According to Approach0 and this search, it is new to MSE. The Details: The torsion-subgroup of an ...
Shaun's user avatar
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6 votes
1 answer
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Let $G$ be a torsion-free abelian group, prove that for $g\neq h\in G$ there exists a homomorphism $\phi:G\to\Bbb{R}$ such that $\phi(g)\neq\phi(h)$

I have the following question: Let $G$ be a torsion-free abelian group, prove that for every distinct elements $g, h \in G$ there exists a homomorphism $\phi: G \rightarrow \mathbb{R}$ such that $\...
Odi's user avatar
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Using a particular theorem, characterise abelian groups which have only finitely many elements of each order (including $\infty$)

This is part of Exercise 4.3.5 of Robinson's "A Course in the Theory of Groups (Second Edition)". According to Approach0 and this search, it is new to MSE. Here is the previous part: An ...
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