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Questions tagged [free-abelian-group]

This is for questions about abelian groups, each with a basis.

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Very basic Question regarding quotients of free abelian groups [closed]

I recently stumbled upon this something that I found a little bit confusing. So say we have some finite free abelian group B of rank m, so we have an expression of B as the direct sum $B = \mathbb{Z}...
froitmi's user avatar
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Question regarding Lang's construction of Grothendieck group

I know there is this post: Question about construction of The Grothendieck group. but it does not answer my question. So we have a commutative monoid M and we then look at the free abelian group $F_{...
froitmi's user avatar
  • 87
2 votes
1 answer
86 views

Direct sum of free abelian group and quotient of abelian group by subgroup

I'm currently studying abelian groups in Kurosh's The Theory of Groups. I'm trying to understand the proof of the theorem: Let $B \leqq A$ be abelian groups. If $A/B \cong C$ and $C$ is a free group, ...
MathematicallyUnsound's user avatar
1 vote
1 answer
92 views

what should be the group $B$?

Here is the exact sequence of abelian groups I am studying: $$0 \to \mathbb Z/2 \to B \to \mathbb Z/2 \to 0 $$ Can I say that $B \cong \mathbb Z/2$ or $B \cong \mathbb Z/2 \oplus \mathbb Z/2$? Is $B \...
Emptymind's user avatar
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2 votes
0 answers
71 views

Seeking an elementry theorem about lattices.

I am doing some work with objects. Each object has a corresponding embedded free $\mathbb Z$-module, with important properties of the object being related to whether the embedding is a lattice. From ...
Sriotchilism O'Zaic's user avatar
9 votes
1 answer
187 views

Is $\text{Hom}(A,\mathbb{Z})$ a product of free abelian groups for all abelian groups $A$?

Let $A$ be an abelian group, and consider the abelian group $\text{Hom}(A,\mathbb{Z})$ of homomorphisms from $A$ to $\mathbb{Z}$. What can be said about this group? Since $\mathbb{Z}$ is torsion-free, ...
Lukas Lewark's user avatar
1 vote
1 answer
176 views

Subgroups of $\mathbb{Z}^n$ isomorphic to $\mathbb{Z}^n$

I am trying to prove the statement that all subgroups of $\mathbb{Z}^n$ isomorphic to $\mathbb{Z}^n$ are of the form $$b_1\mathbb{Z}e_1 \oplus b_2\mathbb{Z}e_2 \cdots \oplus b_n\mathbb{Z}e_n$$ where $...
Hanging Pawns's user avatar
1 vote
1 answer
38 views

Understanding the rank of a cokernel of a free abelian group homomorphism

I am not really familiar with the theory of free $\mathbb Z$-modules so I would appreciate some help understanding this. Let $f: \mathbb Z^3 \to \mathbb Z^3$ be the homomorphism given by the matrix $$\...
nomadicmathematician's user avatar
0 votes
0 answers
72 views

Please check my example of a free abelian group that has the same rank as its subgroup

Is the following correct? The infinite group of integers $\Bbb Z$ under the operation of addition is a free abelian group with generator $1$. The subgroup $2\Bbb Z$ is also cyclic (with generator $2$) ...
Rich C's user avatar
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20 votes
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Is this group free abelian?

Let $K$ be the subgroup of $\mathbb{Z}^\mathbb{Z}$ consisting of those functions $f : \mathbb{Z} \to \mathbb{Z}$ with finite image. Is $K$ free abelian? My guess is no, because $K$ feels too much like ...
diracdeltafunk's user avatar
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0 answers
57 views

Hatcher Simplicial homology [duplicate]

Im trying to solve a Problem from Hatcher: Compute the simplicial homology groups of the $\Delta$-complex obtained from n+1 simplices $\Delta_0^2,\Delta_1^2,...,\Delta_n^2$ by identifying all three ...
NoIdea's user avatar
  • 65
1 vote
1 answer
68 views

A complex of free abelian groups and its homology

Let $L=\{d_i: L_i \rightarrow L_{i-1}\}$ be a complex of free abelian groups. $H(L)=\{H_p(L) \}$ is the homology group of $L$. Then $H(L)$ can be regarded as a complex with differentials zero. I see ...
Xiaosong Peng's user avatar
2 votes
1 answer
58 views

Isomorphism between free abelian group and infinite cyclic group generated by identity

I am currently working through John M. Lee's textbook Introduction to Topological Manifolds but have come across a question that has confused me a little. The exercise is below: Exercise 9.16. Prove ...
JDoe2's user avatar
  • 766
8 votes
1 answer
238 views

Does every finitely presented group have a finite index subgroup with free abelianisation?

Let $G$ be a finitely presented group. Does there exist a finite index subgroup $H$ such that its abelianisation $H^{\text{ab}} = H/[H, H]$ is free abelian? Note, if $G^{\text{ab}}$ is not already ...
Michael Albanese's user avatar
2 votes
1 answer
73 views

The maximal free abelian subgroup that can be embedded in $GL(n,\mathbb{Z})$

I am stuck on this problem and cannot seem to find a good reason for drawing the required conclusion. The problem is as follows: Given $SL(n, \mathbb{Z})$ a subroup in $GL(n, \mathbb{Z})$. How can ...
Yushi MuGiwara's user avatar
1 vote
0 answers
34 views

Splitting of an extension

Let $G$ be a group which is the extension of a free abelian group $A$ of finite rank by a finite simple group $S$. Does $G$ splits over $A$? (that is, $G=F\ltimes A$ for some finite subgroup $F\simeq ...
W4cc0's user avatar
  • 4,160
0 votes
0 answers
48 views

Reference on group-presentations

I'm looking for an introduction to free abelian groups and their presentation, specifically to understand notation such as: $G=^{ab}\left\langle a,b,c \text{ } | a + 2b + 3c = −a − 2b − 7c = 0 \right\...
iki's user avatar
  • 223
0 votes
1 answer
62 views

Partition of minimal generating set [closed]

Let $G$ be a finitely generated abelian group with a minimal generating set $S$. Is it possible to find a partition of $S$ to $\{A,B\}$ such that $A$ generates a free abelian subgroup, and $ B $ ...
David's user avatar
  • 21
2 votes
1 answer
138 views

Does there exist a non-singular matrix $Q \in \mathbb Q^{n\times n}$ with the following two properties?

I am trying to find a non-singular matrix $Q \in \mathbb Q^{n\times n}$ such that: The characteristic polynomials $char(Q)$ and $char(Q^{-1})$ both have some non-integer coefficients. The $\mathbb{Z}...
ghc1997's user avatar
  • 1,641
0 votes
1 answer
51 views

Given a non-singular rational matrix $Q \in \mathbb Q^{n\times n}$, what is $Q(\mathbb Z^{n}) \cap \mathbb Z^{n}$?

Question: Given a non-singular rational matrix $Q \in \mathbb Q^{n\times n}$, is there any necessary condition such that $Q(\mathbb Z^{n}) \supseteq \mathbb Z^{n}$. My thoughts so far: When Q is an ...
ghc1997's user avatar
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3 votes
1 answer
144 views

Proof verification of decomposition of abelian group into torsion and free groups

Claim: Let $A$ be a (not necessarily f.g.) abelian group. Suppose that $T$ is the subgroup of all torsion elements in $A$, and we have that: $${A}\diagup{T} \cong F$$ where $F$ is a free abelian group....
legionwhale's user avatar
  • 2,466
1 vote
0 answers
55 views

Construction of Free abelian groups on Massey Book

I am reading the book of Massey of Algebraic topology, and I am having trouble to understand this construction. Let $ S = \left\{ x_i : i\in I \right\}$. For each index $i$, let $S_i$ denote the ...
Horned Sphere's user avatar
2 votes
0 answers
28 views

Dual group of the torus [duplicate]

Let $T^n=(S^1)^n$ be the torus. We can consider the dual group $\widehat{T^n}=Hom(T^n, \Bbb{C}^*)=\{\phi:T^n\to \Bbb{C}^* : \phi \text{ a group homomorphism}\}$ then $\widehat{T^n}\cong \Bbb{Z}^n$. ...
Chanel Rose's user avatar
0 votes
1 answer
93 views

Obtaining an isomorphism from a surjective homomorphism between abelian groups

Let $f: A\rightarrow B$ be a surjective homomorphism between abelian groups. I want to find such subgroup $B'\subset A$ that $\phi=f|_{B'}:B'\rightarrow B$ would be an isomorphism. I think it's easy ...
Kubrick's user avatar
  • 332
1 vote
1 answer
177 views

Homomorphisms of abelian varieties constitute a finitely generated abelian group

Let $X, Y$ be abelian varieties over $k$. Let $l$ be a prime not equal to the characteristic of $k$. Then one shows that $\text{Hom}_k(X, Y)\to \text{Hom}_{\mathbb{Z}_l}(T_l X, T_l Y)$ is injective. ...
Fabio Neugebauer's user avatar
1 vote
0 answers
85 views

Significance of, and relation between: (1) the Fundamental Theorem of Finitely Generated Abelian Groups and (2) Free Abelian Groups

I'm trying to get the 'big picture' of when each of these might come in handy. So far, it seems that we would prefer to be able to use the Fundamental Theorem of Finitely Generated Abelian Groups (...
Anon's user avatar
  • 1,791
4 votes
0 answers
69 views

Subgroup of free abelian group is free abelian via transfinite induction

There are two common proofs of "if $G$ is free abelian then any subgroup $H$ is also free abelian" when $G$ has infinite rank, one using Zorn's lemma as in Lang's Algebra, one using a well ...
183orbco3's user avatar
  • 1,461
1 vote
1 answer
96 views

Questions about definition of free abelian group [closed]

My professor didn’t define what is a free abelian group $G$ (on a set $X$) but I can guess out the definition. We can define a free abelian group $G$ on a set $X$ as a free object on $X$ in the ...
Sam Wong's user avatar
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0 votes
0 answers
61 views

Understanding "formal sum" in free abelian groups

Despite reading about formal sums and especially the last comment in this post (which seems most relevant to my question) - I still feel the need to make sure I'm not missing something: If there is a ...
Anon's user avatar
  • 1,791
1 vote
1 answer
76 views

Non-Trivial Problem: Determine, up to isomorphism, the following abelian groups:

I'm practicing for an upcoming exam and I have the following exercise: Determine, up to isomorphism, all the abelian groups $A$ that satisfies the 3 following conditions: $A$ has a subgroup $B$ with ...
Jose M Barrientos's user avatar
0 votes
1 answer
64 views

What are the torsion coefficients of $\Bbb Z_{30}\oplus \Bbb Z_{18}\oplus\Bbb Z_{75}?$ [closed]

What are the torsion coefficients of $$\Bbb Z_{30}\oplus \Bbb Z_{18}\oplus\Bbb Z_{75}?$$ I know that $\mathbb{Z}_n \oplus \mathbb{Z}_m \cong \mathbb{Z}_{n\times m} $ iff $\gcd(n,m)=1$, I've tried to ...
Giacomo Davide's user avatar
0 votes
3 answers
282 views

Abelianization of free groups

I'm reading Hatcher's Algebraic Topology and I have some questions about an argument on Page 42: The abelianization of a free group is a free abelian group with basis the same set of generators, so ...
Dasheng Wang's user avatar
2 votes
1 answer
110 views

How to compute rank of free abelian group quotient subgroup generated by 3 elements

Suppose that I have the free abelian group $\mathbb{Z}^3$, and I consider a subgroup $H$ which is generated by the elements $(2, 4, 6), (4, 5, 6), (0,4,8)$. I am trying to compute the rank of the ...
Oderus's user avatar
  • 571
2 votes
1 answer
94 views

Does a group of infinite abelian rank necessarily have exponential growth?

The abelian rank of a group $G$ is the maximum $n$ such that $G$ contains an isomorphic copy of $\mathbb{Z}^n$. A group has infinite abelian rank if it contains $\mathbb{Z}^n$ for every $n$. If $G$ is ...
Rob's user avatar
  • 7,297
0 votes
1 answer
146 views

Express $\mathbb{Z}^2/B$ as a direct product of cyclic groups

In previous questions, I've proved that $B$ is a free abelian group of rank $2$. Then naturally $B$ is isomorphic to $\Bbb Z^2$, right? Then $\mathbb{Z}^2/B$ is isomorphic to $\mathbb{Z}^2/\mathbb{Z}^...
everwith's user avatar
0 votes
1 answer
64 views

Find an homomorphism $\phi:\mathbb{Z}^4 \to G$

I read about free abelian groups would be grateful for an example. Let's say I have to find an homomorphism in this case: Let $G=\langle g\rangle$ be a cyclic (and abelian) group, such that $|G|=4$. ...
Algo's user avatar
  • 2,322
0 votes
0 answers
56 views

Basis of subgroups with full rank of free abelian groups.

Let $A$ be a free abelian group of rank $k$ and $v_1,\dots,v_k$ be its basis. Let $B \le A$ be a subgroup of $A$ of full rank. I want to construct a basis of $B$ from $v_1,\dots,v_k$. One way I can ...
No One's user avatar
  • 8,049
0 votes
0 answers
40 views

Showing that the additive group on the dyadic rationals is not a free Abelian group [duplicate]

I'm wondering whether $(\mathbb{D}, +)$ the additive group of the dyadic rationals is isomorphic to a free Abelian group or not. On an intuitive level, it seems like it's obviously nonfree, but I'm ...
Greg Nisbet's user avatar
  • 11.9k
5 votes
1 answer
144 views

There are only finitely many integer lattices with bounded covolume

I have an uninsightful proof for the following lemma (I discuss motivation below). Let $C>0$ and $n\geq 1$ be fixed, set $\def\Z{\mathbb{Z}}M=\Z^n$. Lemma. There are only finitely many subgroups $...
Olivier Bégassat's user avatar
0 votes
0 answers
49 views

Let $\mathbb{Z}^n,\mathbb{Z}^m$ be a free abelian group, $m<n.$ Prove $\mathbb{Z}^n \not\cong \mathbb{Z}^m$ [duplicate]

Let $\mathbb{Z}^n,\mathbb{Z}^m$ be a free abelian group, $m<n.$ Prove $\mathbb{Z}^n \not\cong \mathbb{Z}^m$ Is it possible that the problem is trivial since $\mathbb{Z}^n$ has $n$ generators and $\...
Algo's user avatar
  • 2,322
3 votes
1 answer
78 views

Let $\mathbb{Z}^n, n\in\mathbb{N}$ be a free abelian group. Prove the intersection of index $h$ subgroups is the subgroup $(h\mathbb{Z})^n$.

Let $\mathbb{Z}^n, n\in\mathbb{N}$ be a free abelian group. Prove the intersection of index $h$ subgroups is the subgroup $(h\mathbb{Z})^n \space \forall h\in \mathbb{N}$ I think I have to prove that $...
Algo's user avatar
  • 2,322
2 votes
1 answer
174 views

Set of homomorphisms on a free abelian group is a free abelian group.

If $G$ is a free abelian group with rank $n$, I need to show that ${\rm Hom}(G,\mathbb{Z})$, set of all homomorphisms is also free abelian group of rank $n$, My work: Since $G$ is free abelian group ...
Adam_math's user avatar
  • 319
0 votes
1 answer
104 views

Abelian group with surjective pushfoward maps and decompsotion of free abelian group

Let $A,B,C$ be abelian groups. $A$ has the property that for any homomorphism $\alpha:A\to B$ and surjective homomorphism $\phi:C\to B$ there exists a homomorphism $\beta:A\to C$ such that $\alpha = \...
Gentleman_Narwhal's user avatar
1 vote
1 answer
241 views

Show that the free abelian group is a group.

Let S be a set and let $F\langle S\rangle = \{\phi : S \to \mathbb{Z}\mid \phi(x) = 0\ \text{ for all but finitely many } x \in S\}$. Show that $F\langle S\rangle$ is an abelian group w.r.t. the ...
user18680's user avatar
0 votes
2 answers
61 views

Uniquness and existence of free abelian group and construction

Let $I$ be a set. A pair $(G,\epsilon)$ consisting of a abelian group $G$ and a map $\epsilon:I \to G$ is called a free abelian group over $I$ if and only if for all abelian groups $H$ and maps $\phi :...
Bill's user avatar
  • 4,503
2 votes
2 answers
219 views

Free abelian group generated by a set

Let $S$ be a set. Then one can construct the free abelian group with basis $S$ as the set of all functions from $S \to \mathbb{Z}$ with finite support: $$\mathbb{Z}^{(S)}:=\{f:S \to \mathbb{Z} \ | \ \...
MaxH's user avatar
  • 389
6 votes
2 answers
175 views

Proof that any element of a free abelian group which is not divisible by any $k>1$ can be extended to a basis

Let $A$ be a free abelian group of rank $n$, and let $\alpha_1 \in A\setminus \{0\}$ such that $\alpha_1 \not \in kA$ for all $k > 1$. Do there always exist $\alpha_2,\ldots, \alpha_n \in A$ such ...
Sebastian Monnet's user avatar
0 votes
0 answers
57 views

Order of a quotient of a free abelian group

Let $G\subseteq \mathbb{C}$ be a free abelian group of rank $n$ and let $p$ be a prime. Then, we know that $$|G/pG|=p^n$$ and in fact for this we don't even need $p$ to be a prime. Suppose now that ...
MarkG99's user avatar
2 votes
1 answer
294 views

Semidirect product of free abelian groups

Consider the semidirect product $$ G=\mathbb{Z}^n \rtimes \mathbb{Z}^m $$ Is $ G $ always virtually abelian? Is it the case that the abelianization of $ G $ is $ \mathbb{Z}^{n+m} $ if and only if $ G $...
Ian Gershon Teixeira's user avatar
3 votes
1 answer
240 views

What is the rank of a subgroup $H$ of finite index $e$ of a free abelian group $G$ of rank $n$?

I've recently been thinking, as it was an implicit corollary to a result in our elementary Abstract Algebra course, about the fact that every subgroup of a free abelian group is also free abelian, ...
Isky Mathews's user avatar
  • 3,285