Questions tagged [abelian-groups]

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How far is a countably infinite reduced abelian $p$-group from being an infinite direct sum?

Question Let $G$ be a countably infinite reduced abelian $p$-group. Is it always possible to write it has an infinite direct sums of non-trivial groups? If it is not true, how far is $G$ from being an ...
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1 vote
1 answer
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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 ...
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  • 256
15 votes
2 answers
718 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 ...
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  • 32.5k
1 vote
1 answer
86 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 ...
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  • 3,271
0 votes
1 answer
84 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 ...
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6 votes
2 answers
226 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 ...
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0 votes
0 answers
59 views

Almost exactness - terminology

Consider a sequence of (torsion-free) abelian groups \begin{equation*} \dots\stackrel{p}{\longrightarrow} G\stackrel{q}{\longrightarrow}\dots \end{equation*} with the property \begin{equation*} \text{...
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10 votes
1 answer
771 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 ...
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11 votes
4 answers
685 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 ...
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7 votes
1 answer
314 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 ...
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0 votes
1 answer
118 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 ...
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  • 505
3 votes
1 answer
213 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 ...
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  • 183
-2 votes
1 answer
128 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)\...
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5 votes
1 answer
293 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 ...
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  • 1,519
2 votes
0 answers
42 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-...
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8 votes
2 answers
413 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 ...
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3 votes
0 answers
127 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 $G^{\infty}$ ...
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  • 341
9 votes
2 answers
252 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 $p^n$. ...
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1 vote
1 answer
236 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 ...
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  • 181
1 vote
1 answer
206 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 ...
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4 votes
0 answers
198 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 ...
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  • 49.5k
8 votes
1 answer
224 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 ...
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  • 49.5k
5 votes
0 answers
154 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 ...
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  • 49.5k
3 votes
0 answers
311 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( ...
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  • 239
4 votes
1 answer
271 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,...
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9 votes
2 answers
527 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, ...
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6 votes
1 answer
336 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 \...
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  • 49.5k
6 votes
0 answers
104 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 ...
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4 votes
1 answer
341 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 \...
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4 votes
1 answer
410 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?
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2 votes
1 answer
151 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 ...
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2 votes
0 answers
60 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,...
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10 votes
2 answers
937 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 $...
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7 votes
0 answers
110 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 ...
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5 votes
1 answer
303 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 $M \in \text{GL}_n(\mathbb{Z})$ ...
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18 votes
0 answers
618 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 ...
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3 votes
1 answer
173 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 ...
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  • 31
3 votes
0 answers
67 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 ...
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  • 2,840
0 votes
1 answer
380 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 ...
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1 vote
1 answer
106 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 ...
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  • 85
8 votes
0 answers
291 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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6 votes
0 answers
249 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 $\bigoplus_{i \in I} D(i,i)$ ...
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1 vote
1 answer
269 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{...
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6 votes
0 answers
140 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 ...
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  • 21.4k
5 votes
0 answers
116 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 ...
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  • 5,626
2 votes
1 answer
213 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}...
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  • 185
7 votes
1 answer
254 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?
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  • 155
4 votes
0 answers
135 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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  • 3,325
0 votes
0 answers
89 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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  • 3,104
1 vote
0 answers
62 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 $n$-...
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