# 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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### 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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### 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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### 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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### 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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### 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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