# Tagged Questions

Should be used with the (group-theory) tag. A group $(G,*)$ is said to be abelian if $a*b=b*a$ for all $a,b\in G.$

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### Is there a strategy for expressing finitely generated abelian group as the direct sum of cyclic groups?

I know that every finitely generated abelian group can be expressed as a direct sum of cyclic groups. I wondering how easily we can find the cyclic groups given an abelian group. Specifically, one of ...
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### Problem from Marcus' Number Fields

I have been stuck on the 27th problem of the 2nd chapter from Marcus' Number Fields. In it we're given $G$, a free abelian group of rank $n$ and its subgroup $H$, which is again of rank $n$. Now, ...
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### Homomorphisms $\frac{\mathbb{Q}}{\mathbb{Z}} \longrightarrow \mathbb{Q}$

Can someone please show me a concrete example of a group homomorphism $$\frac{\mathbb{Q}}{\mathbb{Z}} \longrightarrow \mathbb{Q},$$ if it exists? I apparently cannot find any of it (except for the ...
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### Defining Presheaves on Categories

I'm learning about sheaves and sheaf cohomology with the eventually goal of using these tools to study Riemann surfaces. My references are Forster's Lectures on Riemann Surfaces and Iverson's ...
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### Pontryagin Dual of a Finite Abelian Group [closed]

Let $M$ be a finite abelian group. I want to show that the Pontryagin dual is a finite abelian group, and in particular I am interested in computing the elementary divisors/invariant factors of it. ...
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### Show that $G$ is Abelian if and only if $f: G\times G \to G$ is a homomorphism.

Let $G$ be a group. Let $H$=$G\times G$ be the direct product of $G$ with itself. Define $f: H\to G$ to be $f((g,h))=gh$ for any $(g,h)\in H$. Show that $G$ is Abelian if and only if $f$ is a ...
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### Let A be a finite group and P be a normal p sylow subgroup. What is the connection between P and $Tor_p(A)$

Let A be a finite group and P be a normal p sylow subgroup. can there be an element $g \in A$ where $order(g) = p^x$ where x>0 and $g \notin P$ ? what I really try to understand is the connection ...
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### What triple “tensor product” is this? Is it just isomorphic to a double tensor product?

Consider the abelian groups $A = \Bbb{Q}^{\times}, B = \Bbb{Q}^{\times}, C = \Bbb{Z}^+$. What if we formed a product like: $A \star B \star C = \text{Free}_{\Bbb{Z}}(A \times B \times C)$ ...
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### What is $\mathbb Z \oplus \mathbb Z / \langle (2,2) \rangle$ isomorphic to?

This question came up after I'd solved the following exercise: Determine the order of $\mathbb Z \oplus \mathbb Z / \langle (2,2) \rangle$. Is the group cyclic? I had no trouble solving the ...
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### What is $\mathbb Z^2/\text{Im}(\phi)$ isomorphic to in the following case?

Let $\phi:\mathbb Z^2\to\mathbb Z^2$ be the map $(x,y)\mapsto (x+y,2y)$. I need to find $\mathbb Z^2/\text{Im}(\phi)$. My guess is that this is isomorphic to $\mathbb Z_2$ but I am having ...
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### The center of a group is an abelian subgroup

Let $(G,\circ)$ be a group and let $Z(G):=\{x \in G : ax=xa \ \forall \ a \in G\}$ be the center of $G$. How can I show that $Z(G)$ is an abelian subgroup of $G$? What I did so far: $Z(G)$ is a ...
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### Submodules of a module with a given property

I am curious about the submodules of a module with a given property. Let $M$ be an $R$-module. If $M$ is a finitely generated are the submodules of $M$ finitely generated? If $R=\mathbb Z$, $M$ ...
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### Number of Nonisomorphic Subgroups of Finite Abelian Group

Lets say I have an abelian group $G$ with order $n$ and I am given the primary components of $G$ and their type. How can I determine how many nonisomorphic subgroups of $G$ there are? And as an ...
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### Let G be a finite Abelian group of order $p^nm$, where p is a prime that does not divide m. Then $G=H\times K$ where H and K are the following sets.

I'm trying to follow this proof in my textbook, Contemporary Abstract Algebra by Gallian (p231) but I'm having trouble understanding what's going on. He writes Let G be a finite Abelian group ...
The statement is the following: Given an abelian group $G=\langle a_1,...,a_t\rangle$, and a subgroup $H$ of $G$, we need at most $t$ elements to generate $H$; i.e. $H=\langle b_1,...,b_t\rangle$ for ...