# Questions tagged [abelian-groups]

The abelian-groups tag has no usage guidance.

149
questions

**0**

votes

**0**answers

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**

votes

**0**answers

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 ...

**0**

votes

**1**answer

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**

votes

**0**answers

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**

votes

**0**answers

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**

votes

**1**answer

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**

votes

**1**answer

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**

votes

**0**answers

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 ...

**1**

vote

**0**answers

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 ...

**1**

vote

**0**answers

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**

votes

**1**answer

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**

votes

**1**answer

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**

votes

**2**answers

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**

votes

**1**answer

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**

votes

**1**answer

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**

votes

**1**answer

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**

votes

**1**answer

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**

vote

**1**answer

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**

vote

**1**answer

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**

votes

**0**answers

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**

votes

**1**answer

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**

votes

**4**answers

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**

votes

**2**answers

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**

vote

**1**answer

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**

votes

**1**answer

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**

votes

**2**answers

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**

votes

**0**answers

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**

votes

**1**answer

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**

votes

**1**answer

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**

vote

**1**answer

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**

votes

**1**answer

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**

votes

**0**answers

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**

votes

**0**answers

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**

vote

**0**answers

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**

votes

**0**answers

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**

votes

**2**answers

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**

votes

**1**answer

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**

votes

**2**answers

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**

votes

**0**answers

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**

votes

**0**answers

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**

votes

**0**answers

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**

votes

**0**answers

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**

votes

**2**answers

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**

votes

**1**answer

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**

votes

**1**answer

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**

votes

**0**answers

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**

votes

**1**answer

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

**1**answer

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**

votes

**1**answer

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

**1**answer

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 ...