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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22 views

Rank of the quotient of an Abelian group by its torsion part?

Let $G$ be an Abelian group, and let $G_T$ be the torsion part of $G$. Then my question is, does the rank of $G$ always equal the rank of the quotient group $\frac{G}{G_T}$? Or can they differ in ...
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Bound on the coordinates of torsion points on the elliptic curve: $y^2 = x^3 - k^2x + k^3$

Consider the elliptic curve $y^2 = x^3 - k^2x + k^3$ where $k$ is a non-zero integer. I've come across a question that asks to show that if $(x,y)$ is a rational torsion point on the elliptic curve, ...
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Is an group determined by its torsion part and torsion-free part?

Let $G$ be an abelian group. Let $G_T$ be the torsion part of $G$, i.e. the set of all elements of $G$ of finite order. And let $G_F$ be the torsion-free part of $G$, i.e. the set containing $0$ ...
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A group is called a torsion group if all its elements have finite order [closed]

A group $G$ is called a torsion group if all its elements have finite order. If the $G$'s only finite order element is identity, so $G$ is said to be free of torsion. Let $G$ an abelian group and $T$ ...
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Counting p-torsion elements of an abelian group

Let $A$ be a finite abelian group, and denote by $A_p$ the $p$-torsion part of $A$. Then we have that $|A_p| = |{\rm Surj}(A, \mathbb{Z}/p\mathbb{Z})| + 1$. I came across a proof of this fact. ...
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$fv=0$ group morphisms such $Ker(v),Coker(v)$ are torsion groups, then there is $v'$ such $v'f=0$ and $Ker(v'),Coker(v')$ are torsion groups.

For abelian groups, let $f:G_{1} \to G_{2}$ and $v:G' \to G_{1}$ abelian groups morphisms such as $fv=0$ and $Ker(v), Coker(v)$ are torsion groups. I want to prove the existence of a morphism $v':G_{2}...
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$0 \to G' \xrightarrow{\alpha} G \xrightarrow{\beta} G'' \to 0$ exact sequence, $G$ is torsion group if and only if $G'$ and $G''$ are torsion groups.

Let's work in abelian groups. We say a group $G$ is a torsion group if for every $g \in G$ there is a $n \geq 0$ such as $ng=0$. I want to prove that for an exact sequence $$0 \to G' \xrightarrow{\...
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Torsion in the first Homology group

I was trying to solve an exercise which says the following: Let $U\subset\mathbb{R}^3$ be an open subset. Then $H_1(U;\mathbb{Z})$ has no torsion. I think that the universal coefficient theorem in ...
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If $Z(G)<G$ and $\forall a\in G\setminus Z(G)$, $|a|<\infty$, then $G$ is periodic.

This is Exercise 2.6 of Roman's, "Fundamentals of Group Theory: An Advanced Approach". Searches in Approach0 were unsuccessful due to too many mathematical terms and a search on MSE for &...
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Finding $\operatorname{Tor}(G)$ for the Heisenberg group.

The Heisenberg group $G$ over the field $k$ is the subgroup of $GL_{3}(k)$ defined by the matrices of the form $$ \begin{pmatrix} 1 & x & z\\ 0 & 1 & y \\ 0 & 0 & 1 \end{...
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Show that $z \notin Tor(\mathbb{C^*}).$

I want to answer part(b) here in this question: Let $C^{*}$ denote the group of nonzero complex numbers under multiplication, and $S^{1} \subset C^{*}$ the subgroup of complex numbers of length one. ...
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68 views

A simple reason that $\text{Tor}(\mathbb{C}^{*}) \neq S^1.$

Let $\mathbb{C}^{*}$ denote the group of nonzero complex numbers under multiplication, and $S^{1} \subset \mathbb{C}^{*}$ the subgroup of complex numbers of length one. Torsion elements of $\mathbb{C}^...
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Why the group should be Abelian to say that $(xy)^{mn} = x^{mn} y^{mn}$?

Here is the question I am trying to answer: Show that, if $G$ is Abelian, then the set $Tor(G)$ of torsion elements is a subgroup. Definition: In a group $G, x \in G$ is a torsion element if $x$ is of ...
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51 views

Torsion abelian group and divisible group

If $A$ is a torsion abelian group and $D$ is a divisible abelian group, show that $A\otimes_{\mathbb{Z}}D= 0$. My solution: since $A\otimes_{_\mathbb{Z}}\mathbb{Z} \cong A, D \cong D\otimes_{_\mathbb{...
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Torsion in geometry v.s. Torsion in topology?

Torsion in geometry: There are meanings of torsion of a curve https://en.wikipedia.org/wiki/Torsion_of_a_curve Torsion tensor in Riemannian geometry https://en.wikipedia.org/wiki/Torsion_tensor ...
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Prove that $\operatorname{Hom}_\mathbb Z(K, T') = 0$ for $K = \mathbb Q / \mathbb Z$ and $T'$ reduced torsion.

How does one prove that $\operatorname{Hom}_\mathbb Z(K, T') = 0,$ where $K = \mathbb Q / \mathbb Z$ and $T'$ is a torsion whose maximal divisible subgroup is trivial ? Thank you for your help in ...
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Is torsion of an elliptic curve determined by its reductions modulo primes?

A friend of mine has posed the following question to me:$\newcommand{\Q}{\mathbb Q}\newcommand{\F}{\mathbb F}\newcommand{\Z}{\mathbb Z}$ Let $E$ be an elliptic curve over $\Q$. Is it the case that $|...
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62 views

Steps for computing Tor$(\mathbb{Z}, \mathbb{Z}\times\mathbb{Z})$

I'm reviewing algebraic topology, in particular the Kunneth Formula. I can't find online or in my book (by Hatcher) an explanation for how to calculate $\mbox{Tor}(G,H)$ for any two groups. My ...
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Torsion Points over complex Elliptic Curves

Suppose $G$ is an abelian group of a cubic over $\mathbb{C}$. I need to find group order of $G_{n}=\lbrace x \in G| \exists n \in \mathbb{N}: nx=0 \rbrace$. I solved the problem for $n=2$: $\mathbb{Z}...
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What does the abelianization mean?

Abelianise each of: (a) $\Bbb Q \times S_4$ (b) $D_{12} \times A_4$ (c) $G \times Z_{10}$, where $G$ is the dicyclic group of order 12 and write down the torsion coefficients of the resulting ...
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Clarification on the process of finding torsion coefficients (invariant factors)

I think I'm doing this correctly but I wanted to check. I'm trying to find the torsion coefficients - sometimes called the invariant factors - of the group Z/9 * Z/14 * Z/6 * Z/16. I start by ...
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How to show $\operatorname{coker }(i_{n-1}^\star) \cong \operatorname{Tor}(H_{n-1}(C))$ if $H_{n-1}(C)$ is finitely generated abelian group?

I am studying the Hatcher's Algebraic topology, in the cohomology part (p:195) and I am trying to figure out how to show $\operatorname{coker}(i_{n-1}^\star) \cong\operatorname{Tor}(H_{n-1}(C))$ if $...
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45 views

Applying the elementary divisor theorem

I've just started studying this topic and I've stopped at this exercise: "Let $M = \mathbb{Z}^3$ and $N$ the submodule generated by $\{(1,1,6),(1,-1,6)\}$. Determine a basis of $\{v_1,v_2,v_3\}$ of $M$...
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Trivial intersection of torsion of a module an annihilator of that torsion

I was studying Introduction to Module Theory from the book Dummit D., Foote R. Abstract algebra (3ed., Wiley, 2004). A question related to this topic I got from a friend of mine: Let $A$ be an ...
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171 views

On torsion submodules

Let $A$ be an integral domain and $M$ be a finitely generated $A$-module. Denote by $M_{\mathrm{tors}}$ the torsion sub-module of $M$ and $I$ be the annihilator of $M_{\mathrm{tors}}$. Is it true that ...
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Applying the fundamental theorem of finitely generated abelian groups to the group $\Bbb Z^3/((1,0,3),(-1,2,1))$

Consider two vectors $v_1=(1,0,3), v_2=(-1,2,1)$ in $\Bbb Z^3$. Let $A$ be the subgroup generated by $v_1,v_2.$ Then $\Bbb Z^3/A$ would be a finitely generated abelian group, so by the fundamental ...
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Let $F_n$ be free on $n$ letters and $g_1,…,g_{2m}\in F_n$. Can $F_n/⟨⟨[g_1,g_2],…,[g_{2m-1},g_{2m}]⟩⟩$ have torsion elements?

Let $F_n$ be the free group on $n$ letters. Let $g_1,...,g_{2m} \in F_n$, can the group $$F_n / \langle\langle[g_1,g_2],...,[g_{2m-1},g_{2m}]\rangle\rangle$$ ever have torsion elements? The double ...
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Decomposing a module into a direct sum of torsion and free part?

I know by the fundamental theorem of finitely generated abelian groups that we can write any $\mathbb{Z}$-module of the form $$A\cong \mathbb{Z}^r\oplus Tor(A) $$ Can we not also decompose any ...
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A group satisfying the minimal condition on subgroups is a torsion group.

Definition: A group $G$ has the minimal condition on subgroups (m.c.s.) if every descending chain $G_1>G_2>...$ in $G$ is finite. Let $G$ have the m.c.s. and suppose $G$ is not torsion. If $G$ ...
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About the tensor product of two abelian torsion-free groups.

(I) Let $L^*=L \otimes \mathbb{Q}$ where $L$ is an abelian torsion-free group and $\mathbb{Q}$ is the additive group of rationals. Since $L$ is torsion-free the mapping $g \mapsto g \otimes 1$ is a ...
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Do Elements of Odd Order Form a Subgroup

I was looking at this question on the odd torsion points of elliptic curves. The accepted answer states that the "coprime-to-$p$" points form a subgroup (so, in particular, the $C_{oddtors}(\mathbb{Q})...
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Torsion group of $y^2 = x^3 \pm nx$

Let $n\in \mathbb{Z}$ be not divisible by $15$ and suppose $n$ and $-n$ are both not perfect squares. Prove that at least one of the elliptic curves $Y^2 = X^3 + nX$ and $Y^2 = X^3 - nX$ has torsion ...
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Is there an established concept slightly weaker than torsion, as defined like this?

Let $\sim$ be an equivalence relation over an algebraic object $X$ whereby there is a modular algebra $X/{\sim},\star$. Does there already exist a concept ever so slightly weaker than torsion, ...
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If the torsion subset of a group has “finite index”, is the group torsion?

Let $T=\mathrm{Tor}(G)$ be the torsion subset of $G$, i.e. $$T=\mathrm{Tor}(G):=\{g\in G: g^n=1 \text{ for some } n\geq 1\}.$$ In general $T$ is not a subgroup of $G$, so it doesn't make sense to talk ...
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Subgroups of the Torsion Subgroup of an Elliptic Curve

A silly question. I am working through Silverman and Tate, Rational Points on Elliptic Curves Let $E$ be an elliptic curve given by a Weierstrass equation $$E:y^2=x^3+ax^2+bx+c, \quad a,b,c \in \...
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If $M=D/(d)$ then $M=t_{p}(M)\oplus t_{q}(M)$

I am interested in the following result: Let $D$ be a $PID$. Let $d\in D$ such that $d=p^{r}q^{s}$ with $s,r\geq 1$ where $p,q$ are non-associated irreducible elements. If $M=D/(d)$ then $M=t_{p}(M)...
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Is $\lvert 3x+2^{\nu_2(x)}\rvert\leq\lvert x\rvert$ in the monoid quotient $\Bbb N^+/\langle3,4\rangle$?

What does the monoid quotient $\Bbb N^+/\langle3,4\rangle$ look like, where $\Bbb N^+$ is the multiplicative monoid produced by multiplying primes? In particular, is it by any chance relatively quick ...
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Let $H$ be a group, if its abelianization is $\mathbb{Z}_2$ does this mean that $H$ has torsion?

As is mentioned in the title, I have some group $H$ and I know that its abelianization is $\mathbb{Z}_2$. Does this imply that $H$ has torsion? Edit: Since people want more context, here's some ...
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1answer
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Free torsion part of centralizer of a cyclic group [closed]

Suppose $G$ is a cyclic group $C_{5} <S_{5}$. Does anybody know what the torsion-free part of the centralizer of this group is?
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Torsion of simple abelian group extension

I'm curious if there is a general way to determine the torsion of a group extension. I'm most interested in the simple example where we have a central extension $$ 1 \to \mathbb Z \xrightarrow{f} G \...
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Hands-on definition of cotorsion quotient

Let $M$ be an $R$-module for some ring $R$. We usually define the torsion submodule $\mathrm TM\subseteq M$ as $\mathrm TM=\{m\in M\colon \exists r\in R, r\nmid 0\colon rm=0 \}$. We call $M$ torsion ...
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class of abelian groups

What class of abelian gropus $\cal K$ is this a very natural one: the members of $\cal K$ are of the following form: the direct sum $\oplus$ of finitely many $\mathbb Z_n$' s and finitely many $\...
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1answer
57 views

Every finitely generated module of a PID is a direct sum of a free module and of a torsion module

Let $R$ be a PID. Show that every finitely generated $R$-module is a direct sum of a torsion module and of a free module. Attempt: There's a theorem that claims that if $M$ is a finitely generated $...
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1answer
56 views

Why $\mathbb{Q}$-vector space has no $\mathbb{Q}$-torsion?

Why $\mathbb{Q}$-vector space has no $\mathbb{Q}$-torsion ? What does the notion of $\mathbb{Q}$-torsion technically mean ?
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how is the target of $\text{tor}_n^R(A,\ )$ related to $\mathbf{Ab}$?

I do not follow in the snippet below (taken from Advanced Modern Algebra, Rotman, Vol. 2,page 277,3rd edition) how it is meant that the target of $$\text{tor}_n^R(A,\ )$$ may be smaller or larger then ...
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1answer
154 views

elliptic curve $~y^2 = x^3 + 3x +8~ (mod~ 13)~$ - torsion groups

Corollary $6.4.$ Let $E$ be an elliptic curve and let $m \in \mathbb{Z}$ with $m \neq 0$. $(b)$ If $m \neq 0$ in $K$, i.e., if either $\operatorname{char}(K) = 0$ or $p := \operatorname{char}(K) > ...
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1answer
42 views

Is the following relation correct?

Using Kunneth theorem I would like to compute the following 4th homology: $ H_4(S^2\times S^3/\mathbb{Z}_k,\mathbb{Z})\simeq H_4(S^2,\mathbb{Z})\otimes H_0(S^3/\mathbb{Z}_k,\mathbb{Z})\oplus H_3(S^2,\...
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1answer
46 views

What can we say about the homomorphism restricted to this subgroup?

Let $G$ be a finitely generated abelian group. So, we know that $G$ is isomorphic to $$\mathbb{Z}^d \oplus \mathbb{Z}_{q_1} \oplus \cdots \oplus \mathbb{Z}_{q_n}$$ for some $d \geq 0$ and powers of ...
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2answers
49 views

Surjective implies injective in this case?

Let $G$ be a torsion free, finitely generated abelian group. Suppose that $f : G \to G$ is an epimorphism (surjective homomorphism). Does it follow that $f$ is an isomorphism? My thoughts: this is ...
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2answers
79 views

Commutator power lying in commutator subgroup

The argument used in the proof of Proposition 3 of this math.SE answer appears to prove the following claim: Let $G$ be a group and let $H\subseteq G$ be a normal subgroup. Let $n\geq 0$ and let $x,...