All Questions
Tagged with abelian-groups short-exact-sequences
8 questions
12
votes
2
answers
2k
views
Do all exact sequences $0 \rightarrow A \rightarrow A \oplus B \rightarrow B \rightarrow 0$ split for finitely generated abelian groups?
Suppose $A$ and $B$ are finitely generated Abelian groups. Are all exact sequences of the form $0 \rightarrow A \rightarrow A \oplus B \rightarrow B \rightarrow 0$ split?
If not, is there an example?
- 307
8
votes
2
answers
501
views
Exact sequence of $n$th powers of abelian groups
Let $A,B,C$ be finitely generated abelian groups. Assume that there is an exact sequence $$0 \to C \to A^n \to B^n \to 0,$$where $A^n = A \oplus \dotsc \oplus A$ as usual. It is not assumed that $A^n \...
- 63.1k
8
votes
1
answer
393
views
Pontryagin dual of a group-cohomology class
Let $A, B, C$ be finite Abelian groups fitting in a short exact sequence
$$
1 \rightarrow A\overset{\iota}{\rightarrow} B\overset{\pi}{\rightarrow} C\rightarrow 1
$$
This determines a class $[\...
- 813
4
votes
1
answer
242
views
Quadratic refinements of a bilinear form on finite abelian groups
$\DeclareMathOperator\Hom{Hom}$Let $A$ be a finite abelian group and $\text{Sym}(A)$ the (abelian) group of symmetric bilinear forms over $A$ valued in $\mathbb{R}/\mathbb{Z}$.
A quadratic function on ...
- 813
4
votes
1
answer
497
views
Short exact sequence $0\to \mathbb Z\to A \to \mathbb R \to 0$
Does every short exact sequence $0\to \mathbb Z\to A \to \mathbb R \to 0$ split in the category of Abelian groups?
- 51
4
votes
0
answers
212
views
When does a short exact sequence of abelian groups with $B\cong A\oplus C$ split?
$\hspace{20pt}$Duplicate on stackexchange.
This question, in a way, extends this one. The question is what are some sufficient conditions on the abelian group $B$ so that if $B\cong A\oplus C$ and a ...
- 171
3
votes
2
answers
305
views
Are $H^3(A,U(1))$ and $\operatorname{Ext}^1(A,A^\vee)$ isomorphic for $A$ finite Abelian?
Motivated by three-dimensional Dijkgraaf-Witten TQFTs for finite Abelian groups $A$, that are classified by $H^3(A,\mathbb{R}/\mathbb{Z})$, it seems natural that this group is (naturally) isomorphic ...
- 813
2
votes
3
answers
1k
views
Finite / uniquely divisible abelian groups
Is there any counter example for the following statement?
STATEMENT:
Let $0 \to F \to A \to Q \to 0$ be a short exact sequence of abelian groups.
Assume that $F$ is a finite group, and $Q$ is a ...
- 945