# Questions tagged [free-abelian-group]

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### Prove that a surjective homomorphism between $\mathbb{Z}^3$ to itself is isomorphism.

I am trying to prove theat a surjective homomorphism between $\mathbb{Z}^3$ to itself is actually an isomorphism. The question was given after the material of free abelian groups was taught. I ...
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### Every torsion free abelian group is free abelian?

There is a theorem in Lang's Algebra, that if $G$ is a "finitely-generated" torsion-free abelian groups are free abelian. (More generally, I saw in wiki that every finitely generated torsion-free $R$-...
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### The counit of an adjunction of the adjoint pair free and underlying functor

I would like to know how the counit $\varepsilon$ for an adjunction $(F,U,\phi)$ $$\varepsilon:FUX\to X$$ works if $F$ is the free functor from $\mathbf{Set}$ to $\mathbf{Ab}$ and $U$ is the ...
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### I get a contradiction in the theory of free abelian groups. What am I doing wrong?

Hi: The definition I'll use is this: Let $F$ be an abelian group and $X$ a subset of $F$. Then $F$ is a free abelian group on $X$ if for every abelian group $G$ and every function $f$ from $X$ to $G$ ...
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### How to alternatively prove: Any subgroup of any free Abelian group is free

I am a bit hesitant to ask this question, but here goes. We call an Abelian group $\textit{free}$ if it has some basis $\{e_i\mid i\in I\neq\emptyset\}$. Unlike vector spaces, not all (non-trivial) ...
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### Proving that the following subset of $\mathbb{Z}$ is the subgroup generated by its minimum

Let $F$ be a free abelian group, i.e. $F \simeq \mathbb{Z}^n$ for some $n \in \mathbb{N}$, $n \neq 0$ and let $G < F$ be a subgroup. Consider the following subset of $\mathbb{Z}$: \begin{equation} ...
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### Question about the definition of a free abelian group

I am kind of lost when I try to understand the definition of the free abelian group on a set $X$ I read the question here What is the definition of a free abelian group According to this question, ...
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### how to deduce two matrices $P$ and $Q$ are square if $PQ=I_{n}$ and $QP=I_{m}$

If you had two matrices $P$ and $Q$ where $P$ is an $n\times m$ matrix and $Q$ is an $m \times n$ matrix both with integer entries satisfying: $$PQ = I_{n} \text{ and } QP = I_{m}$$ How would you ...
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### Universal Property…The Map is not Well-defined?

How does the following proposition from this book make sense: Proposition 2.8 Let $G$ be the group defined by the presentation $(X,R)$. For any group $H$ and map of sets $\alpha : X \to H$ sending ...
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### Proving the Free Abelian Group is Free Abelian…?

On page 40 of these notes is the following exercise: Prove that the group with generators $a_1,...,a_n$ and relations $[a_i,a_j]=1$, $i \neq j$, is the free abelian group on $a_1,...,a_n$. On ...
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### Does sufficient information on the multiplicative group of the fraction field of a GCD and Factorization domain, captures unique factorization?

Let $R$ be (non-field) an Atomic (https://en.wikipedia.org/wiki/Atomic_domain) and a GCD domain (https://en.wikipedia.org/wiki/GCD_domain) of characteristic zero. Let $U(R)$ denote the multiplicative ...
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### What could be the elementary divisors of subgroup of $\mathbb{Z}^2$

What can be the elementary divisors of subgroup $H \le \mathbb{Z}^2$ of index $36$? I can't see what's the connection between the index and the elementary divisors? As far as I know, elementary ...
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### Elementary divisors of a subgroup of $\mathbb{Z^2}$ of index $36$

The question is what might be the elementary divisors of a subgroup $H\leq \mathbb{Z^2}$ of index 36? I need to list all the possible options. My solution: The theorem states that if $A$ is an ...
### How many epimorphisms from $F_2$ to $\mathbb{Z}_5$?
How many epimorphisms are there from $F_2$ (the free group with $2$ generators) to $\mathbb{Z}_5$? If $F_2$ is generated by $2$ elements, so it has a basis of rank$2$, meaning that it is isomorphic ...
### Amount of elements of order $p^2$ [closed]
Let $p \neq 5$ be a prime number. How many elements of order $p^2$ are there in $\mathbb{Z_p} \times \mathbb{Z}_{p^5} \times \mathbb{Z}_{25}$? I have no idea how to even approach this... Any hints?