Questions tagged [abelian-groups]

For questions about groups whose elements commute.

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Eigenvalue function of the representation variety of free abelian group

$\DeclareMathOperator\SL{SL}$Let $\rho:\mathbb Z^n\rightarrow \SL(n,\mathbb C)$ be a representation of a finitely generated free abelian group, by simultaneously triangularization, we can assume that $...
0 votes
0 answers
67 views

countable direct sum of cyclic abelian $p^{2}$ groups

Let $G={{\Bbb{Z}}_{p^{2}}}^{(\aleph)}$ (countable direct sum of copies of ${\Bbb{Z}}_{p^2}$). It is clear that every subgroup of $G$ is a homomorphic image of $G$. Now this is my question: Is it true ...
4 votes
1 answer
489 views

Why does the category of abelian groups satisfy the axiom AB6?

In his Tohoku article, Grothendieck says that the category $\mathbf{Ab}$ of abelian groups satisfies the axiom AB6, namely "All small colimits exist in $\mathbf{Ab}$. Moreover for any index ...
1 vote
1 answer
86 views

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 ...
15 votes
4 answers
1k views

Zero-sum partition of an abelian group

This is a question I have been asking myself some 5 years ago. I later got bored by lack of progress, but maybe some additive combinatorialists here know further. I'm not claiming it is conceptual or ...
14 votes
2 answers
776 views

Groupoid cardinality of the class of abelian p-groups

$\DeclareMathOperator\Aut{Aut}\newcommand\card[1]{\lvert#1\rvert}$So, after going over the classification of finite abelian groups in a class I was teaching this winter, I got curious about whether it ...
1 vote
1 answer
154 views

Number of orbits for abelian group actions

Suppose $G$ is an abelian group acting faithfully on two sets, $X$ and $Y$, of the same size. None of $G$, $X$ and $Y$ is finite. Now suppose $G$ is the union of abelian groups $G_i$, where $i$ varies ...
0 votes
1 answer
118 views

Any abelian Lie subgroup containes a connected Lie subgroup of codimension 1 [closed]

I am trying to understand the proof of the following claim (see A.L. Onishchik, E.B. Vinberg (Eds.) Lie Groups and Lie Algebras III, p.50, Theorem 3.1). Theorem 3.1 (ii) If the Lie group $G$ is ...
6 votes
2 answers
339 views

Abelian groups such that $A \cong \mathrm{End}(A)$ and "complete rings"

Motivation: for any ring $R$ there is the natural monomorphism $\mathrm{in} \colon R \to \mathrm{End}(R_{add}): r \mapsto (x \mapsto rx)$, where $R_{add}$ is an additive abelian group ( rings are ...
11 votes
1 answer
1k views

Are condensed vector spaces over finite fields always solid?

The Clausen-Scholze theory of condensed mathematics offers an abelian category with enough projective objects that embraces the study of arbitrary locally compact (and Hausdorff) groups. The behaviour ...
12 votes
4 answers
1k views

How many non-isomorphic abelian subgroups of the permutation group $S_n$?

I am interested in how many (pairwise non-isomorphic) subgroups of the permutation group $S_n$ are abelian. ($n \in \mathbb{N}$ arbitrary and possibly big) Are you aware of any references which treat ...
2 votes
2 answers
4k views

Any factor group of a finite abelian group is isomorphic to some subgroup

If you visit this link, you'll see at the top of the PDF view. Basic properties of finite abelian groups: Every quotient group of a finite abelian group is isomorphic to a subgroup. If the above ...
7 votes
1 answer
413 views

Existence of abelian group extension relative to group homomorphism

Let $f: A \to B\ $ be an abelian group homomorphism. Are there abelian groups $G,\ H,\ K$ such that $K \subseteq H \subseteq G$ and the map $$\pi \circ i: H \to G/K$$ which is the composition of ...
6 votes
4 answers
2k views

Is a torsion-free abelian group finitely generated, if all of its localizations at primes $p$ are finitely generated over $\mathbb{Z}_p$?

Background: When proving that the group of $k$-isogenies $\mathrm{Hom}_k(A,B)$ between two abelian varieties is finitely generated, one first shows that the Tate map $$\mathbb{Z}_\ell\otimes_{\mathbb{...
3 votes
0 answers
327 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 $\mathrm{SL}_{2}$; i.e. $V:=\left\{\left( ...
0 votes
1 answer
174 views

Fourier transform on lattice strip

I am working with a triangular lattice $L=\{n_1 a_2 + n_2 a_2 : n\in\mathbb{Z}^2 \}$ and $a_1 = \pmatrix{1 \\ 0}$ and $a_2 = \frac{1}{2} \pmatrix{-1 \\ \sqrt{3}}$, and I want to compute the Pontryagin ...
8 votes
0 answers
1k views

Examples of uncountable abelian $p$-groups

Does anyone know of any interesting examples of an infinite abelian $p$-group which is uncountable? By non-interesting here I mean the direct sums of cylic and quasi-cylic groups, and totally ...
5 votes
2 answers
386 views

Size of distinct sums in A

Let $G$ be an abelian group. Let $A\subset G$ be a finite set. $\sum_A$ is defined as: $$\left\{\sum_{b\in B}b \mid B\subset A\right\}$$ Is there any result similar to Freiman's Theorem for $\sum_A$? ...
70 votes
9 answers
16k views

Is there a slick proof of the classification of finitely generated abelian groups?

One the proofs that I've never felt very happy with is the classification of finitely generated abelian groups (which says an abelian group is basically uniquely the sum of cyclic groups of orders $...
3 votes
1 answer
411 views

Structures of subgroups of a finite abelian p-group

$\newcommand\la{\langle}\newcommand\ra{\rangle}$Let $G=\mathbb{Z}/p^{i_1}\times\cdots\times\mathbb{Z}/p^{i_r}$ with $i_1\leq\ldots\leq i_r$ be a finite abelian $p$-group. Then there can be many ...
-2 votes
1 answer
156 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)\...
7 votes
1 answer
468 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 ...
5 votes
1 answer
304 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 ...
6 votes
0 answers
283 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
1 answer
324 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 ...
8 votes
2 answers
480 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 ...
5 votes
1 answer
157 views

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 votes
0 answers
137 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 $G^{\infty}$ ...
10 votes
1 answer
311 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
1 answer
261 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 ...
6 votes
0 answers
314 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 ...
11 votes
1 answer
441 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
0 answers
190 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 ...
4 votes
1 answer
366 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,...
10 votes
2 answers
745 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
1 answer
415 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 \...
4 votes
1 answer
386 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
1 answer
476 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?
4 votes
1 answer
198 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
0 answers
65 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
2 answers
1k 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 $...
7 votes
0 answers
116 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 ...
6 votes
1 answer
318 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 $M \in \text{GL}_n(\mathbb{Z})$ ...
3 votes
1 answer
182 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 ...
0 votes
1 answer
518 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 ...
3 votes
0 answers
86 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 ...
1 vote
1 answer
123 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 ...
10 votes
1 answer
2k views

A group whose automorphism group is cyclic

Is there an Abelian group $A$ which is not locally cyclic whose automorphism group is cyclic ? This question was first posted here.
15 votes
1 answer
534 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 \...
8 votes
0 answers
307 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 ...