Questions tagged [abelian-groups]

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votes
1answer
120 views

Are isomorphic quotients of abelian groups induced by automorphisms? [closed]

If I have an (abelian) group $G$ and an automorphism $\sigma: G \to G$ then for any subgroup $H$ of $G$ there is an induced isomorphism $G/H \cong G/\sigma(H)$ given by the map $gH \mapsto \sigma(g)\...
5
votes
1answer
262 views

Quotient groups obtained by quotienting $G^n$ by $G^{n-1}$

Notation: For a group $G$, we write $G^n$ to denote the $n$-fold direct product of $G$ with itself. Problem set up: Consider, for a finite group $G$, and $n > 1$, the set $Q(G)_n$ of all ...
2
votes
0answers
38 views

Cotorsion-freeness in uncountable products of abelian groups

An abelian group $A$ is cotorsion provided that whenever $A \leq G$ with $G$ abelian and $G/A$ is torsion-free, we have $G \cong A \oplus B$ for some $B \leq G$. An abelian group $A$ is cotorsion-...
8
votes
2answers
361 views

On $p$-groups with abelian automorphism group

Let $G$ be a $p$-group of order $p^{n}\geq p^{7}$ and its automorphism group is elementary abelian $p$-group. Then, it is clear that $G$ is nilpotent of class $2$. However, the converse is not true in ...
3
votes
0answers
122 views

The group of sequences in $G^{\mathbb{N}}$ that converge to $(0,0,0,\dots)$

Let $G$ be a discrete abelian group and $G^{\mathbb{N}}$ be the direct product (with the product topology), which consists of sequences $(a_1,a_2,a_3,\dots)$ of elements of $G$. Let $G^{\infty}$ ...
9
votes
2answers
190 views

Do these properties of a countable abelian group guarantee a Prüfer subgroup?

Suppose $(G,+)$ is a countable abelian group and $p$ is a prime number such that: The subgroup $pG$ has finite index in $G$, and For every $n \in \mathbb{N}$, $G$ contains an element of order $p^n$. ...
1
vote
1answer
210 views

Bound for order of a group depending on number of elements of maximal order

This question has been partly answered in MSE, see here. In a paper On the Number of Elements of maximal order in a Group, it is proven that an arbitrary group $G$ with a finite number of elements of ...
1
vote
1answer
186 views

How many elements of each order are there in this $p$-group? [closed]

Let $G$ be a countable Abelian $p$-group which equals a direct sum of at most countably many finite cyclic groups and at most countably many copies of the Prüfer $p$-group, where these finite cyclic ...
3
votes
0answers
157 views

Uncountable Mittag-Leffler condition?

Let $(X_\alpha)_{\alpha <\kappa}$ be an inverse system of abelian groups. If $\kappa = \omega$ (or by extension if $\kappa$ is of countable cofinality), then the Mittag-Leffler condition is a ...
8
votes
1answer
207 views

Example of an uncountable sequence of abelian groups with nonvanishing $\varprojlim^2$?

$\DeclareMathOperator{\op}{\mathrm{op}}\DeclareMathOperator{\Ab}{\mathsf{Ab}}\DeclareMathOperator{\Vect}{\mathsf{Vect}}$Question 1: What is an example of a sequence $(X_\alpha)_{\alpha<\kappa}$ of ...
5
votes
0answers
139 views

Can an infinite abelian $p$-group be tall and thin?

Does there exist an abelian $p$-group $A$ with countable Ulm invariants and uncountable height? Here by height, I mean the minimal ordinal $\rho$ such that $p^\rho A$ is divisible [1]. For an ordinal ...
2
votes
0answers
167 views

Homology $H_{\ast}(T, V)$

Let $A$ be a local domain. We let $T=T(A) $ be the subgroup of $\mathrm{SL}_{2}$ consisting of diagonal matrices and $V$ be the subgroup of unital matrices of $\mathrm{SL}_{2}$. We have that $T$ acts ...
4
votes
1answer
210 views

A question on bi-character of finite abelian group

Setting: $G$ is a finite abelian group and any bicharacter on $G$, where a bi-character on $G$ is a map $b:G \times G \to \mathbb{Q}/\mathbb{Z}$ such that $$b(x+y,z)=b(x,z)+b(y,z),b(x,z+y)=b(x,z)+b(x,...
9
votes
2answers
431 views

Is there a non-degenerate quadratic form on every finite abelian group?

Let $G$ be a finite abelian group. A quadratic form on $G$ is a map $q: G \to \mathbb{C}^*$ such that $q(g) = q(g^{-1})$ and the symmetric function $b(g,h):= \frac{q(gh)}{q(g)q(h)}$ is a bicharacter, ...
6
votes
1answer
298 views

Which abelian groups are $\varprojlim^1$ groups?

Question 1: Let $\mathcal A$ be an abelian group. Does there exist an inverse system $(A^n)_{n \in \mathbb N} = (\cdots \to A^n \to A^{n-1} \to \cdots \to A^0)$ such that $\varprojlim^1 A^\bullet \...
6
votes
0answers
90 views

Cohomology of the Baer-Specker group

Let $A = \prod_{i \in \mathbb{N}} \mathbb{Z}$ be the Baer-Specker group; that is, a countably infinite product of the integers. We will consider this as a discrete abelian group. Are the higher ...
4
votes
1answer
321 views

The action of the unitary divisors group on the set of divisors and odd perfect numbers

Let $n$ be a natural number. Let $U_n = \{d \in \mathbb{N}\mid d\mid n \text{ and } \gcd(d,n/d)=1 \}$ be the set of unitary divisors, $D_n$ be the set of divisors and $S_n=\{d \in \mathbb{N}\mid d^2 \...
4
votes
1answer
357 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?
1
vote
1answer
132 views

Sum of divisors and unitary divisors as the eigenvalue and the spectral norm of some addition matrix?

Let $n$ be a natural number and $D_n$ be the set of divisors. We can make this set to a ring by observing that each divisor $d$ has $$0 \le v_p(d) \le v_p(n)$$ Hence we can add two divisors $d,e$ by ...
2
votes
0answers
55 views

Alternating $n$-homomorphism on abelian group is skew of $n$-cocycle

Let $A$ be a finitely generated abelian group. Let $c$ be a 2-cocycle on $A$, where $A$ acts trivially on $\mathbb{C}^\times$. It is well-known that the skew-map $$ c(a_1,a_2) \longmapsto \frac{c(a_1,...
10
votes
2answers
755 views

Classification of subgroups of finitely generated abelian groups

A finitely generated abelian group $A$ is isomorphic to a direct sum of cyclic groups. I am interested in an extension of this result on couples of abelian groups $(A,B),$ where $B$ is a subgroup of $...
2
votes
0answers
86 views

Who classified varieties that are commutative groups?

Who are the authors of the theorems asserting that connected varieties/manifolds which are abelian groups are isomorphic to ${\bf R}^k \times {\bf T}^n$? In the smooth setting, I guess this is due to ...
7
votes
0answers
107 views

A conjecture on circular permutations of n elements in an abelian group of odd order

In 2013 I formulated the following conjecture in additive combinatorics. Conjecture. Let $G$ be an additive abelian group of odd order, and let $A$ be a subset of $G$ with $|A|=n>2$. Then, there is ...
5
votes
1answer
293 views

Can all proper sublattices of $\mathbb{Z}^n$ be generated cyclically?

Let $\Lambda \subset \mathbb{Z}^n$ be a proper sublattice (so that $\Lambda \ne \mathbb{Z}^n$). We say that $\Lambda$ is cyclically generated if there exists a matrix $M \in \text{GL}_n(\mathbb{Z})$ ...
18
votes
0answers
532 views

If $A, B$ are abelian groups such that $\mathrm{Hom}(A, G) \cong \mathrm{Hom}(B, G)$ for all abelian groups $G$, must $A$ and $B$ be isomorphic?

The question is in the title. If the isomorphism $\mathrm{Hom}(A, G) \cong \mathrm{Hom}(B, G)$ is natural in $G$ then this is just the Yoneda Lemma. If $A$ and $B$ are finitely generated this is also ...
3
votes
1answer
170 views

Can one turn finite-dimensional vector subspaces into a cancellative semigroup?

Let $V$ be a vector space over some field and let ${\rm Fin}\,V$ be the family of all finite-dimensional subspaces of $V$. Is it possible to turn ${\rm Fin}\,V$ into an commutative cancellative ...
3
votes
0answers
63 views

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
votes
2answers
342 views

Abelian groups and their subgroups

It is well known that every finite abelian group is a direct product of cyclic groups. So for every $n$ every finite abelian group of exponent $n$ is a direct product of cyclic groups of order at most ...
1
vote
1answer
102 views

Irreducible non-Abelian subgroup of $\mathrm{U}_n(\mathbb{C})$, containing diagonal matrices

Consider an irreducible non-Abelian subgroup $\mathrm{H}$ of group of unitary matrices $\mathrm{U}_n(\mathbb{C})$, that contains the subgroup of diagonal matrices. Does there exist any result ...
8
votes
0answers
280 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 ...
6
votes
0answers
237 views

When is every element of a coend of abelian groups contained in one of the summands?

Let $I$ be a small category and let $D : I^{\mathrm{op}} \times I \to \mathsf{Ab}$ be a functor. The coend $$\int^{i \in I} D(i,i)$$ can be constructed as the direct sum $\bigoplus_{i \in I} D(i,i)$ ...
1
vote
1answer
202 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
0answers
138 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 ...
5
votes
0answers
98 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
1answer
200 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
1answer
248 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
0answers
130 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 ...
0
votes
0answers
86 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
0answers
60 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
1answer
302 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
1answer
241 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
2answers
376 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
1answer
243 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
1answer
133 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
1answer
226 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
1answer
593 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
1answer
161 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
1answer
141 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
0answers
122 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
1answer
334 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 $\leftrightarrow$ ...