All Questions
Tagged with abelian-groups reference-request
26 questions
1
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1
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209
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A question about automorphism group of abelian group
Does anyone know any references that describe automorphism group $\operatorname{Aut}(\mathbb R^n\times \mathbb T^m)$? I searched for a long time but couldn't find it.
1
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0
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134
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Isomorphic quotients of a countably infinitely-generated free abelian group
Let $F$ denote the free abelian group on countably infinite generators. I am trying to understand the relationship between normal subgroups $A$ and $B$ of $F$ with isomorphic quotients. So is there a ...
1
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0
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64
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Groups with prescribed Ulm invariants
In Kaplansky's book infinite abelian groups he provides (through some exercises) a complete classification of $p^{\infty}$-torsion countable abelian groups in terms of Ulm invariants. In other words ...
2
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0
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408
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Conceptual proof of fundamental theorem of finite abelian groups
I'm looking for a conceptual proof of the following statement:
Lemma: Let $G$ be a finite abelian $p$-group. Let $a$ be an element of maximal order. Then $G=\langle a \rangle \times H$ for some ...
4
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0
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113
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Duality for finite quotient groups of finitely generated free abelian groups
$\newcommand{\Z}{{\Bbb Z}}
\newcommand{\Q}{{\Bbb Q}}
\newcommand{\Hom}{{\rm Hom}}
$ The following lemma is certainly known.
Lemma (well-known).
Let $B$ be a lattice (that is, a finitely generated ...
1
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0
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97
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A duality of finite groups coming from a surjective homomorphism with finite kernel of algebraic tori
$\newcommand{\Hom}{{\rm Hom}}
\newcommand{\Gm}{{{\mathbb G}_{m,{\Bbb C}}}}
\newcommand{\X}{{\sf X}}
$ I am looking for a reference for the following lemma (for which I know a proof):
Lemma.
Let $\...
1
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1
answer
89
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Element of order $p$ and finite height $\geq1$ in a reduced abelian group $p$-group with an element of order $p^2$
This is a reference request for the following statement:
Fact:
Let $G$ be a reduced abelian $p$-group with an element of order $p^2$. Then $G$ contains an element of order $p$ and of finite height at ...
3
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0
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98
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Hales' generalization of the stacked bases theorem (seeking a proof)
In his paper Analogues of the stacked bases theorem, published in the proceedings of a 1976 conference, A.W. Hales claimed some interesting generalizations of the stacked bases theorem for abelian ...
0
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0
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94
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Name for a pair of lattices one of which having theta series with coefficients a subsequence of another lattice's theta series coefficients
Is there a name for a pair of lattices which have the property given in the title (up to a change of variable)? The following example of a pair captures the property mentioned above:
$$(i)\ 1 + 80q^3 ...
6
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1
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276
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Reference request: an elementary result on characters of finite abelian groups
The referee of a paper I submitted to a journal asked me to include a reference of the following elementary result on characters of finite abelian groups:
Let $A$ be a finite abelian group of order $...
2
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1
answer
190
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Logic article on first-order invariants of abelian groups
I remember reading an article published in the 1970s by a Polish mathematician describing the first-order invariants of a torsion-free abelian group. I do not recollect the author's name, the title of ...
2
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0
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96
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A kind of cancellation ; exchange problem for groups
For which $(m,n,k,l) \in (\mathbb N\cup \{0\})^4$ , with $m\le n ; k\le l$ , does there exist a group $G$ with a finite subnormal series with torsion-free Abelian quotients such that $G \times \mathbb ...
2
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0
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130
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Existence of a transfinite sequence of abelian groups having a strange property
I am studying a paper which uses the following lemma. The context is irrelevant, as the lemma is only used as a technical trick and has no pointer to a reference or hint in the proof but its link to ...
5
votes
1
answer
163
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Characteristically simple locally compact abelian groups
Say a topological group $G$ is topologically characteristically simple if there does not exist a closed subgroup $1 < K < G$ such that $K$ is invariant under all automorphisms of $G$ (here `...
3
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2
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318
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Character kernels in the lattice of subgroups of a finite abelian group
I am looking for any efforts that have been made to characterize the character kernels (equivalently, the subgroups yielding cyclic quotients) inside the lattice of subgroups of a finite abelian group....
9
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0
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298
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An abstract zero-sum problem
I would like to know whether the problem described below has appeared in the literature and/or whether similar questions have been studied. I would be very happy to find some references or, if none ...
7
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1
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617
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Looking for a modern source about Ulm Invariants
I'm looking for a modern, approachable text (preferably a website, textbook, or expository article, and preferably one easily available online or at a library) which can explain the concept of Ulm ...
2
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1
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159
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Counting elements with certain word length in abelian groups
Given a (finite) abelian group $G = \langle S \mid R \rangle$, has the problem of counting the number of elements which can be expressed as a word (in $S$) of length $\leq k$ been studied? If so, ...
5
votes
1
answer
472
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Countable reduced abelian group containing all countable reduced abelian groups
Is there a countable abelian group for which its subgroups are exactly all of the countable "reduced" abelian groups? (Reduced means that its divisible subgroup is zero)
Is the following group a ...
2
votes
1
answer
194
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Name and references for a "twisted" addition in a ring
This question may seem a little unmotivated, but it actually isn't something that just occurred to me out of nowhere. It stems from a discussion I had the other day with a friend and is directly ...
9
votes
1
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3k
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Automorphism group of a finite group
I would like to ask if there exists an explicit description of $\mathrm{Aut}(G)$, the group of automorphisms of a finite group $G$, in particular, when $G$ is abelian. E.g., if $G = \mathbb{Z}/m\...
6
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2
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647
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The 2-group of extensions
Let $A,B$ objects of an abelian category. Then we can define the abelian group $\mathrm{Ext}^1(A,B)$ as the set of isomorphism classes of extensions $0 \to B \to E \to A \to 0$, endowed with the Baer ...
3
votes
1
answer
767
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Linear algebra of finite abelian groups
If $f: V \to W$ is a surjective homomorphism of vector spaces, and we have fixed a basis for $V$, it is always possible to find a basis for $W$ such that the matrix associated to $\phi$ in the two ...
8
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2
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2k
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Modern reference for integral homology of a finitely generated abelian group
I'm interested in the calculation of $H_n (G,\mathbb{Z})$ for G a finitely generated abelian group, particularly for $n=3$. It's carried out in the 1954/55 Séminaire Henri Cartan, titled "Alg&...
11
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1
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3k
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Where can I easily look up / calculate (abelian) group cohomology?
For an example I'm trying to understand, I need to calculate some cohomology group of some $\mathbb Z$-module with coefficients in some other $\mathbb Z$-module (with no interesting actions). (In ...
9
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3
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3k
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Why are divisible abelian groups important?
I just quote wikipedia:
"Divisible groups are important in understanding the structure of abelian groups, especially because they are the injective abelian groups."
I am asking for detail ...