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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32 views

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 $R$-...
3
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1answer
37 views

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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3answers
103 views

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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0answers
73 views

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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0answers
27 views

surjective homomorphism between free abelian groups and first isomorphism theorem

So i recently started studying homology groups and whenever i stumbled upon its general idea or motivation, i came across computations such as $$ \langle a,b,c\rangle / \langle c \rangle = \langle a,...
3
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1answer
45 views

Given surjective Homomorphism $f:A \to B$ then $A \cong \ker(f) × B$

Given surjective Homomorphism $f:A \to B$ when $B$ is a free abelian group. Prove: $A \cong \ker(f) × B$ I have seen this claim in the following link: Abelian group admitting a surjective ...
3
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1answer
57 views

Finding the order of a coset of a quotient group.

I'm dealing with free abelian groups and I encountered the following question: Let $A$ be an abelian free group with a basis $\{x_1, x_2, x_3\}.$ Let $B$ be subgroup of A generated by $x_1+x_2+4x_3$, ...
2
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1answer
28 views

Given a free abelian group and a basis, prove there exists an automorphism between two elements

I'm studying free abelian groups and still don't have the intuition on how to approach questions. Let $A$ be an abelian free group, and let {$e_1, e_2, e_3$} some basis. Let $a=2e_1+7e_3$ and $b=3e_2+...
3
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1answer
72 views

Finding an Explicit Basis for a Free Abelian Group

Let $A$ be the free abelian group generated by the intervals $[a,b] \subseteq \mathbb{R}$, with $a,b \in{\mathbb{Q}}$. Let $B$ by the group obtained by quotienting out by the relations $[a,b] + [b,c] ...
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0answers
38 views

Galois cohomology of free abelian groups

I am trying to understand the cohomology groups $H^1(\Gamma, \mathbb{Z}^r)$, where $\Gamma$ is a finite (or profinite) group with a continuous action on $\mathbb{Z}^r$ (in my setting, $\Gamma$ will be ...
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1answer
60 views

On free abelian groups

I'm learning about the concept of a free abelian group. First question: nowhere it is stated that these groups cannot be finite, but the definition seems to imply it. Is this true? Second question: ...
2
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1answer
23 views

Any subgroup of free abelian group $\mathbb{Z}^r$ of index $n$ contains subgroup $n\mathbb{Z}^r$.

Any subgroup of index $n$ of free abelian group $G= \mathbb{Z}^r$ contains subgroup $n\mathbb{Z}^r$. My attempt to prove this is as follows: If I have any subgroup $F$ of index $n$ of G, then ...
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0answers
32 views

Showing two simple tensors are in the same coset of the tensor product

I'm reviewing for a practice qualifier I get to take during my first week of grad school. I have reviewed tensors from a module theory perspective and am working on a classic proof to show that the ...
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2answers
69 views

Grothendieck Group of the integers under multiplication

I want to construct the group $\mathbb{Q}\setminus 0$, the nonzero rational numbers under multiplication, from the monoid $\mathbb{Z}\setminus 0$ using the Grothendieck group construction. First I ...
1
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1answer
66 views

Here I prove every free group is free abelian. Where is the mistake?

Let $F$ be free over $X$. Then for any group $G$ and any $\alpha: X \to G$ there is a homomorphism $\beta: F \to G$ such that $\alpha = \beta|X$. Alright. Now, in particular, when $G$ is abelian $F$ ...
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0answers
35 views

What notions, theorems, proofs connect Free Abelian Groups, $F(n)$, factored by a subgroup with eigenvalues and eigenvectors?

Just finished a semester in abstract algebra. We learned about free abelian groups on finite generators and the Smith Normal Form algorithm to find the elementary divisors which describe said free ...
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135 views

Order of the set of group homomorphisms from $\mathbb{Z}^n$ into an arbitrary finite group $G$.

Question: Let a finite $G$ act on itself by conjugation, and let $N$ be the number of conjugacy classes. Find a formula for $|\mathrm{Hom}\left(\mathbb{Z}^n, G\right)|$, denoting the set of group ...
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1answer
44 views

Munkres' Section 67 Exercise 6 — Rank of the subgroups of free abelian group with finite rank.

This is from Munkres' Section 67 Exercise 6, which focuses on proving that Any subgroup $B$ of a free abelian group $A$ with rank $n$ has rank at most $n$. The proof of this statement has been ...
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1answer
42 views

Show that if $G$ is free abelian with basis $\{x, y\}$, show that $\{2x+3y, x+y\}$ is also a basis for $G$

This is from Munkres' Section 67, but there is a typo in the original question which is as follows: Show that if $G$ is free abelian with basis $\{x, y\}$, show that $\{2x+3y, x-y\}$ is also a ...
4
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1answer
37 views

Finding the quotient of this free abelian group

I have the group $\langle a,b,c\rangle/\langle -b+c-a,b+c-a\rangle$. I know this is $\mathbb{Z}\oplus\mathbb{Z_2}$. However, I tried doing it like this and got something else : I have $$-b+c-a=0, b+...
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0answers
18 views

The Smith Normal Form of …

The Problem is : An abelian group $A$ is the cokernel of the map between two free abelian groups defined by the matrix $M$ ; where $M=\left( \begin{array}{ccc} 2 & -2 & 6 \\ 10 & 14 &...
3
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0answers
44 views

Isomorphism between free abelian groups that fit into a short exact sequence

Let $A$ and $B$ be free abelian groups and $0\to A\to B\to\Bbb{Z}/p\to 0$ a short exact sequence. Are $A$ and $B$ necessarily isomorphic? The only examples I can come up with are of the form $...
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1answer
59 views

Quotient of $\Bbb Z^{\Bbb N}$ by a cyclic subgroup is isomorphic to itself

Consider the group $G=\Bbb Z^{(\Bbb N)}$, the direct sum of countably many $\Bbb Z$'s. (Thus $\Bbb Z^{(\Bbb N)}$ is free abelian of rank $| \Bbb N|$.) Let $a$ be any nonzero element of $G$. Then $<...
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0answers
34 views

What is the free abelian group of an infinite set?

Given a set $S$, the free abelian group is defined as the set of formal sums of elements of $S$. Is this restricted to formal sums of a finite subset of elements of $S$? For example, is $S = \mathbb{R}...
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1answer
63 views

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} ...
2
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2answers
79 views

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, ...
2
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1answer
44 views

Invariant basis of some free $\mathbb{Z}$-module

Let $G$ be a group acting on $M:=\mathbb{Z}^{r}$ a $\mathbb{Z}$-module. My question is that, if a submodule $N$ of $M$ is invariant under $G$, does $N$ have to have a basis which is invariant under $...
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2answers
42 views

Basis of $G/H$ for free abelian groups G.

If I have a free abelian group G with basis $\{g_i\}$ and H a subgroup of G. Then H is free abelian and a basis is given by the elements of $\{g_i\}$ that are in $H$, right? Then under what condition ...
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2answers
38 views

$0\to A_1 \xrightarrow{\phi} A_2 \xrightarrow{\psi} A_3 \to 0$, if $A_3$ is free abelian then the sequence splits.

I have a proposition that says that for an exact sequence $0\to A_1 \xrightarrow{\phi} A_2 \xrightarrow{\psi} A_3 \to 0$, if $A_3$ is free abelian then the sequence splits. So to prove this, we ...
2
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1answer
46 views

map $A \otimes B \to A \otimes C$ induced by inclusion an injection even if $A$ not free? [duplicate]

I was reading something in Munkres' book on algebraic topology and at some point I got confused. He sais that if $B$ is a subgroup of $C$ and if $A$ is free, then the map : \begin{equation} A \...
1
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1answer
85 views

Direct sum of $n$ (infinite) cyclic groups isomorphic to direct sum of $n$ copies of $\mathbb{Z}$?

I'm currently selfstudying some algebra and i am currently covering the various equivalent definitions of free abelian groups. However, in order to understand why these definitions are indeed ...
1
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1answer
92 views

How do we show two functions have orthogonal actions on a set or group?

If we take initially $X=\{x\in \Bbb Z[\frac12]\setminus0\}$ then we can see pretty quickly that from any given starting number, the actions of the two functions on $X$ will generate the whole set: $f(...
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2answers
113 views

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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2answers
74 views

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 ...
3
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1answer
58 views

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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1answer
88 views

Basis of subgroup of a free abelian group

We have this theorem: ** Let $F$ be a free abelian group of rank $n$ and let $H$ be a subgroup of $F.$ There exists a basis $\{x_1,...,x_n\}$ of $F$ and integers $d_1,...,d_r > 0 $ such that • $...
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1answer
239 views

An abelian group $G$ is free abelian if and only if satisfies the projective property

I've been reading group theory from Rotman's book "Introduction to the theory of groups" and in Chapter 10 of free abelian groups there is an exercise which am having a hard time to prove. The ...
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0answers
39 views

If $G=A\times B$, where $A$- torsion group, $B$-free abelian group, how to show that $A=G_{tor}$

$G$ is a finitely generated Abelian group with $G=A\times B$ where $A$ is a torsion group and $B$ a free Abelian group. I know that every finitely generated Abelian group can be factored into cyclic ...
2
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2answers
68 views

Let $A$ be an abelian finitely generated free group and $A/B$ be a torsion group. Show that $rank(A)=rank(B)$.

Let $A$ be an abelian free group that is finitely generated, and let $B\subset A$ be a subgroup of $A$ such that $A/B$ is a torsion group. Show that $rank(A)=rank(B)$. From the hypothesis, I know ...
3
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2answers
122 views

Is studying a free group (or other free object) equivalent to considering only the consequences of the basic axioms?

I'm trying to get a better understanding of the rationale behind free groups, and more generally free objects. This answer does a great job at explaining how various free objects are built, and I ...
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0answers
39 views

How prove a group is not free abelian group [duplicate]

Anyone can give me a hint please? "Show $\mathbb{Z}^{\mathbb{N}}$ is not a free-abelian group"
0
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1answer
78 views

Alternative definition on free modules.

I 'm trying to compare Free Modules and Free Abelian Groups. We know that, Definition. An abelian group $G$ is called free abelian group with rank $n\in \Bbb N^*$, if $G$ is the direct sum of $n$ ...
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2answers
47 views

Clarification on what is and is not a free abelian group

I have a question very similar to that asked in Free $\mathbb{Z}$-modules, but is not answered in that thread. This thread cleared up some of my confusion, but still leaves me with the question: Why ...
2
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2answers
88 views

Can $G/2G$ and $\mathbb Z / 2\mathbb Z \times … \times \mathbb Z / 2\mathbb Z$ be not isomorphic? $G$ is a free abelian group with finite basis.

This is from Theorem 67.8 of Munkres Topology: The proof gives stronger than bijective correspondences, specifically isomorphisms between $G$ and $\mathbb Z \times ... \times \mathbb Z$ and $2G$ and $...
4
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1answer
47 views

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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1answer
76 views

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 ...
3
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0answers
116 views

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 ...
-1
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2answers
98 views

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 ...
0
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2answers
80 views

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?