Questions tagged [free-abelian-group]
This is for questions about abelian groups, each with a basis.
196
questions
1
vote
1
answer
45
views
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}...
- 87
0
votes
0
answers
45
views
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_{...
- 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, ...
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 \...
- 2,087
2
votes
0
answers
70
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 ...
- 2,258
9
votes
1
answer
186
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, ...
- 353
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 $...
- 245
1
vote
1
answer
37
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
$$\...
- 5,601
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 $2$) ...
- 35
20
votes
1
answer
915
views
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 ...
- 15.7k
0
votes
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 ...
- 65
1
vote
1
answer
67
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 ...
- 1,628
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 ...
- 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 ...
- 101k
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 ...
