Questions tagged [free-abelian-group]
This is for questions about abelian groups, each with a basis.
204 questions
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Give an example of an abelian group in which the only element of finite order is the identity but it is not a free abelian group. [duplicate]
I am trying to find an example of an abelian group in which the only element of finite order is the identity but it is not a free abelian group. I know that if we add an extra condition that the group ...
1
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1
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Tensor product $\mathbb{Z}^n\otimes \mathbb{Z}^m$
For a problem in my PhD I'm using Leray–Hirsch Theorem (as presented in the Hatcher) to compute a cohomology ring. For a particular group $G$, I have the s.e.s.
$$
\def\Inn{\operatorname{Inn}}
\def\...
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$G$ is a free abelian group, with its rank being $\aleph_0$. [closed]
Say $G$, a free abelian group, has rank $\aleph_0$. We need that $G \oplus \mathbb{Z}\oplus\mathbb{Z} \cong G\oplus\mathbb{Z}$.
So, I understand there is no torsion part, so we have that $G \cong \...
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Proving the Group of Continuous Functions from $\mathbb{N}_{\{\infty\}}\to \mathbb{Z}$ is a Free Abelian Group.
I am taking a course on Topology and Groups and have a question from an exercise sheet which introduces free Abelian groups. We haven't covered these formally yet in the course, so I think the ...
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Quotient of two subgroups of the free abelian group $\mathbb{Z}^r$ and Smith Normal Form.
Suppose we have free abelian groups $H \leq G \leq \mathbb{Z}^r$. I'd like to understand the quotient $G/H$. Suppose $\operatorname{rank}(H)=p$, $\operatorname{rank}(G)=s$ and that $(h_i)_{i=1,..,p}, (...
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Does the dual group of a free abelian group separate subgroups?
Let $F$ be a free abelian group. Let $A,A' \subseteq F$ be two subgroups. Assume that for all $f \in F^* = \mathrm{Hom}(F,\mathbb{Z})$ we have $f(A) = f(A')$. Does this imply $A = A'$?
Equivalently: ...
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Reference request for characteristic subgroups of free abelian groups
The characteristic subgroups of a free abelian group $F$ are the $k F$ with $k \in \mathbb{N}$. (In particular, they are verbal and hence fully invariant.) I know a proof when $F$ has finite rank, but ...
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Intuition Behind Forming a Quotient of a Free Abelian Group by a Single Relation
I'm studying free abelian groups and came across the construction of a quotient group defined by introducing a single relation. Specifically, if $G=H/\langle r\rangle$, where $H$ is a free abelian ...
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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}...
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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_{...
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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, ...
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1
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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 $$
Can I say that $B \cong \mathbb Z/2$ or $B \cong \mathbb Z/2 \oplus \mathbb Z/2$? Is $B \...
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Seeking an elementry theorem about lattices.
I am doing some work with objects. Each object has a corresponding embedded free $\mathbb Z$-module, with important properties of the object being related to whether the embedding is a lattice. From ...
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Is $\text{Hom}(A,\mathbb{Z})$ a product of free abelian groups for all abelian groups $A$?
Let $A$ be an abelian group, and consider the abelian group $\text{Hom}(A,\mathbb{Z})$ of homomorphisms from $A$ to $\mathbb{Z}$. What can be said about this group?
Since $\mathbb{Z}$ is torsion-free, ...
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Subgroups of $\mathbb{Z}^n$ isomorphic to $\mathbb{Z}^n$
I am trying to prove the statement that all subgroups of $\mathbb{Z}^n$ isomorphic to $\mathbb{Z}^n$ are of the form
$$b_1\mathbb{Z}e_1 \oplus b_2\mathbb{Z}e_2 \cdots \oplus b_n\mathbb{Z}e_n$$
where $...
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Understanding the rank of a cokernel of a free abelian group homomorphism
I am not really familiar with the theory of free $\mathbb Z$-modules so I would appreciate some help understanding this.
Let $f: \mathbb Z^3 \to \mathbb Z^3$ be the homomorphism given by the matrix
$$\...
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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 $2$) ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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0
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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 ...
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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\...
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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 $ ...
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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}...
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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 ...
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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....
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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 ...
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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 $\widehat{T^n}\cong \Bbb{Z}^n$. ...
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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 ...
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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. ...
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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 (...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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3
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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 ...
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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 ...
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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 ...
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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}^...
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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 $|G|=4$.
...
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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 ...
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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 ...
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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 $...
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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 $\...
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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 $...
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1
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203
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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 ...
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