Questions tagged [abelian-groups]

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60 views

Fitting subgroup of a solvable group is a direct product of some elementary abelian $p$-groups

I saw a remark saying If $G$ is solvable, then ${\rm Out}(F(G))$ is isomorphic to a direct product of ${\rm GL}(n_i,p_i)$ (except for certain value of $n_i$ and $p_i$). I know the key is to prove ...
7
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0answers
220 views

A $\mathsf{ZF}$ example of a nonreflexive group which is isomorphic to its double dual?

Given a group $G$ denote by $G^\ast=\mathrm{Hom}(G,\Bbb Z)$ its dual and by $j\colon G\to G^{\ast\ast}$ the canonical homomorphism $g\mapsto (f\mapsto f(g))$. A group is reflexive iff $j$ is an ...
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1answer
181 views

Is there a free profinite abelian group on a profinite set?

Let $\mathit{Profinite}_{\mathrm{Ab}}$ be the category of profinite abelian groups, and let $\mathit{Profinite}_{\mathrm{Set}}$ be the category of profinite sets. Does the forgetful functor $$\mathit{...
6
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131 views

When is $\{s_2-s_1,s_3-s_2,s_1-s_3\}\cap S$ non-empty for any $s_1,s_2,s_3\in S$?

A subset $S$ of an abelian group is a subgroup if and only if it is closed under taking differences; that is, the difference of any two elements of $S$ is in $S$. Suppose, however, that we only know ...
4
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0answers
74 views

Invariant measure on coset space and integrable functions

Let $G$ be a locally compact abelian group, and $H$ a closed subgroup. Let $C_c(G)$ be the space of continuous, compactly supported complex valued functions on $G$. General theory of Haar measure ...
2
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1answer
191 views

Abelian torsion-free group with $\mathbb{Z}_2\times\mathbb{Z}$ as automorphism group

Let $A$ be an abelian torsion-free group. If $A$ is isomorphic with the group of rational numbers whose denominators are powers of, say, $2$, then its automorphism group is isomorphic with $\mathbb{Z}...
7
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1answer
228 views

Additive group of local rings

Is there a theory or characterization for those finite $p$-groups that can be considered as the additive group of a finite local commutative ring with identity?
4
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0answers
119 views

Image of $\rm{lim}^1$ functor

In category of abelian groups, we know that — values of $\rm{lim}^1$ on countable systems are precisely cotorsion groups — values of $\rm{lim}^1$ on systems of finitely generated groups are of the ...
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0answers
84 views

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 ...
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0answers
58 views

Pure (ordered) subgroups

Let $H,G$ be abelian groups with $H \leq G$. We say that $H$ is a pure subgroup of $G$ if for every $n \in \mathbb N$ and $h \in H$ the following holds: If $h$ is $n$-divisible in $G$, then $h$ is $n$-...
5
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1answer
283 views

Finite-by-torsion-free abelian groups (or compact abelian groups with finitely many components)

Here's a question I should know the answer to but don't: Suppose $1\to F \to G \to G/F \to 1$ is a short exact sequence of abelian groups with $F$ finite and $G/F$ torsion-free. Must the sequence ...
3
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1answer
231 views

What are the almost periodic functions on the complex plane?

The almost periodic functions on the real line can be characterized as uniform limits of trigonometric functions. I was wondering whether a similar definition exists on the complex plane (a locally ...
4
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2answers
331 views

Non-torsion part of the abelianisation of congruence subgroups

I've posted this question on math.stackexchange, but haven't gotten any responses so I'm trying here instead. Let $A = F_q[T]$ be the ring of polynomials in one variable with coefficients in a finite ...
3
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1answer
209 views

Finite dimensional compact abelian group that is not a product of connected and a totally disconnected

Let $G$ be a compact abelian group. A compact abelian group is said to have dimension $n$ if $\dim_\mathbb{Q} \mathbb{Q}\otimes \hat G = n$. Equivalently one can show that this holds if $G$ is ...
3
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1answer
128 views

On decomposition of finite Abelian groups

It is easy to see that for any finite Abelian group $G$ and any numbers $a,b$ with $|G|=ab$ there exist a subgroup $A\subset G$ and a subset $B\subset G$ such that $|A|=a$, $|B|=b$ and $G=A+B$, where $...
6
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1answer
220 views

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 $...
12
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1answer
500 views

How nearly abelian are nilpotent groups?

It is not uncommon to read that "nilpotent groups are 'close to abelian'."1,2 Can this sentiment be made precise in the sense of the Turán and Erdős definition of "the probability that two elements of ...
1
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1answer
138 views

Inverse limit of $p^n$-torsion abelian groups

Let $p$ be a prime and let $\{A_n\}_{n > 0}$ be an inverse limit of abelian groups such that $A_n$ is $p^n$-torsion with $A_n/p^{n - 1} \cong A_{n - 1}$ (these isomorphisms are part of the data). ...
1
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1answer
113 views

On the laplacian of connected, undirected, multigraphs without loops

Let $G$ be a connected, undirected multigraph, without loops. Let $L_G = D_G - A_G$, where $D_G= diag (val (v_1), \ldots , val (v_n) )$ where $n$ is the no. of vertices of $G$ and $val (v_i)$ ...
5
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0answers
115 views

Which rings are the endomorphisms ring of some abelian groups?

Which rings are (isomorphic to) the endomorphisms ring of some abelian group? Is there any necessary and sufficient condition?
4
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1answer
259 views

What are the LCA groups that are the Pontryagin dual of a locally profinite abelian group?

For certain subcategories of LCA groups, we have nice descriptions of the dual category under Pontryagin duality (all groups are implicitly assumed to be abelian): finite groups $\leftrightarrow$ ...
19
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4answers
1k views

Constructively, is the unit of the “free abelian group” monad on sets injective?

Classically, we can explicitly construct the free Abelian group $\newcommand{\Z}{\mathbb{Z}}\Z[X]$ on a set $X$ as the set of finitely-supported functions $X \to \Z$, and so easily see that the unit ...
5
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2answers
361 views

What is $\mathrm{Hom}(\mathbb{Q},\mathbb{Z}(p^{\infty}))$?

What is $\mathrm{Hom}(\mathbb{Q},\mathbb{Z}(p^{\infty}))$? I have a reference that says the group in question is $\mathbb{Q}_p,$ the additive group of the quotient field of the $p$-adic integers. Can ...
1
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1answer
97 views

Indecomposable monoids

Let $M$ be a commutative reduced and cancellative monoid and $K(M)$ its group of quotients. We say that $M$ is indecomposable if for every divisor-closed submonoids $M_1$ and $M_2$, $M=M_1\oplus M_2$...
0
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1answer
121 views

exact short sequence of divisible groups splits? [closed]

Let $0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$ be a short exact sequences of divisible abelian groups. Does then the sequence splits?
4
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2answers
372 views

Co-finite type abelian groups

Suppose $B$ is an abelian group such that for every integer $n\ge 1$, the $n$-torsion subgroup $B[n]$ is finite. Let $B_{\rm tor} = \varinjlim_{n\ge 1} B[n]$ be the torsion subgroup of $B$. Is it ...
2
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0answers
165 views

Trivial Tate modules

Let $A$ be an abelian group, and $p$ a prime. I'll call $$T_p(A) := \text{Hom}_{\mathbf{Z}}(\mathbf{Q}_{p}/\mathbf{Z}_{p}, A).$$ If $A$ is finite, then $T_p(A)$ is trivial, but the converse is not ...
2
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1answer
135 views

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 ...
6
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1answer
271 views

Transcendence degree of the fraction field of $k[G]$ for torsion free abelian group $G$

Let $k$ be a field of characteristic $p$ and $G$ be a torsion free abelian group . Then the group ring $k[G]$ is an integral domain , let $k(G)$ denote its field of fractions . Then can we say ...
1
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1answer
95 views

Limit of trace maps in finite fields

If $\mathbb{F}_{q^n}$ is a finite field with $q^n$ elements ($q$ being a power of a prime $p$) we have the trace map $tr^n_m:\mathbb{F}_{q^n}\rightarrow \mathbb{F}_{q^m}$ such that $x\mapsto x+F^m(x)+....
3
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1answer
123 views

Characterisation of a class of group homomorphisms related to a central extension

Let $S$ and $R$ be groups and say $\sigma: S \twoheadrightarrow R$ is a group homomorphism that is a central extension; that is, it is surjective (extension) and its kernel is contained in the centre ...
2
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0answers
81 views

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 ...
4
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0answers
104 views

sums of quadratic forms over finite abelian groups

Let $A$ be a finite abelian group. Let $q:A\times A\to \mathbb{C}^{\times}$ be a non-degenerate bicharacter (that is: for every $a\in A$ $q(a,-)$ and $q(-,a)$ are characters of $A$, which are trivial ...
1
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0answers
59 views

Symmetric analogue of “alternating bihomomorphism is skew of 2-cocycle” theorem

Let $G$ be a finite abelian group. It is well-known that every alternating bihomomorphism $\Omega:G\times G \to \mathbb{C}^\times$ arises as the skew $\kappa/\kappa^T$ of a 2-cocycle $\kappa \in Z^2(G,...
5
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0answers
329 views

Subgroups and quotients of an abelian pro-finite group

It is well known that every subgroup $H$ of a finite abelian group $G$ is isomorphic to a quotient of $G$. I'm wondering whether there is a counterpart for profinite groups. For example is it true ...
6
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2answers
177 views

Two abelian groups, each being direct factor of the other

Let $M$ and $N$ be two abelian groups. Suppose that $M$ is a direct factor of $N$ (i.e. there are homomorphisms $i:M\rightarrow N$ and $p:N\rightarrow M$ such that $p\circ i=id_M$) and $N$ is also a ...
14
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1answer
505 views

$\mathbb{Z}$-module structure of the subring generated by an algebraic number

Let $a$ and $b$ be algebraic numbers which are not necessarily algebraic integers. Is there some invariant that allows us to determine whether $\mathbb Z[a]$ and $\mathbb Z[b]$ are isomorphic as $\...
8
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2answers
366 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 \...
3
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0answers
36 views

What is an example of an integral domain with a module that is 1-separable but not separable?

Let R be an integral domain. All modules under discussion are torsion free unital left R-modules.     An R-module is completely decomposable if it is the direct sum of rank 1 submodules.     An R-...
2
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0answers
65 views

Are these convex cones polyhedral?

I'm actually playing with some convex cones, and I would like to know if there is a chance they would be described by a finite number of inequalities. Let me introduce some notation first. Let $n\...
2
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0answers
77 views

Alternating bihomomorphism is skew of 2-cocycle - relative situation

Let $G$ be a finite abelian group. It is well-known that every alternating bihomomorphism $\Omega:G\times G \to \mathbb{C}^\times$ (i.e. $\Omega(g,g)=1$) arises as the skew $\kappa/\kappa^T$ of a 2-...
2
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0answers
120 views

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 ...
4
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2answers
413 views

Number of torsion-free abelian groups

Let $\mathfrak{c}$ be the cardinality of the continuum. How much Choice, if any, is needed to prove that there are $2^{\mathfrak{c}}$ distinct (mutually nonisomorphic) torsion-free abelian groups of ...
14
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1answer
447 views

Torsion-free abelian group $A$ such that $A \not \simeq A \oplus \Bbb Z \simeq A \oplus \Bbb Z^2$

Is there a torsion-free abelian group $A$ such that $A \not \simeq A \oplus \Bbb Z \simeq A \oplus \Bbb Z \oplus \Bbb Z$ (as groups)? Notice that $\Bbb Z$ is not cancellable, so $A \oplus \Bbb Z \...
11
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1answer
375 views

Functorial description of mod-2 homology of an abelian group $A$ in terms of $A/2$ and ${}_2A.$

Let $A$ be an abelian group and $p$ be a prime. If $p\ne 2,$ there is a very nice functorial description of the homology algebra $H_*(A,\mathbb Z/p):$ $$H_*(A,\mathbb Z/p)\cong \Lambda^*(A/p)\otimes \...
26
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0answers
618 views

The field of fractions of the rational group algebra of a torsion free abelian group

Let $G$ be a torsion free abelian group (infinitely generated to get anything interesting). The group algebra $\mathbb{Q}[G]$ is an integral domain. Let $\mathbb{Q}(G)$ be its field of fractions. ...
2
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1answer
136 views

Constructing an explicit extension of a continuous character on a closed subgroup of a certain locally compact abelian group

Let $ G $ be a locally compact abelian group and $ \omega: G \times G \to \mathbb{T} $ a continuous multiplier on $ G $, i.e., \begin{align} \forall r,s,t \in G: \qquad \omega(s,t) ~ \omega(r,s + t) &...
5
votes
1answer
564 views

Direct limits of a matrix and its transpose

Let $A \in M_n(\mathbb Z)$ and $A^T$ denote the transpose of $A$. Define the direct limits $$H_1 = \mathrm{colim} (\mathbb Z^n \xrightarrow{A} \mathbb Z^n \xrightarrow{A} \mathbb Z^n \xrightarrow{A} \...
5
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1answer
288 views

Subgroups of the tensor product $A\otimes A$

I have this problem about subgroups of the tensor product of an abelian group $A$ with itself which arises from a complete different setting. I fell into this question studying quandles and quandle ...
2
votes
1answer
124 views

Intermediate lattices $C\mathbb{Z}^n \subseteq \Lambda \subseteq \mathbb{Z}^n$

Let $C \in \mathfrak{gl}(\mathbb{Z},n)$ be a symmetric full rank integer valued matrix (in my case its the symmetric part of a Cartan matrix).. Let $\Lambda \subseteq \mathbb{Z}^n$ be a full rank ...