# Questions tagged [abelian-groups]

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### 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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### 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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### 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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### 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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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 $... 0answers 86 views ### Who classified varieties that are commutative groups? Who are the authors of the theorems asserting that connected varieties/manifolds which are abelian groups are isomorphic to${\bf R}^k \times {\bf T}^n$? In the smooth setting, I guess this is due to ... 0answers 108 views ### A conjecture on circular permutations of n elements in an abelian group of odd order In 2013 I formulated the following conjecture in additive combinatorics. Conjecture. Let$G$be an additive abelian group of odd order, and let$A$be a subset of$G$with$|A|=n>2$. Then, there is ... 1answer 299 views ### Can all proper sublattices of$\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})$... 0answers 565 views ### If$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$are finitely generated this is also ... 1answer 172 views ### Can one turn finite-dimensional vector subspaces into a cancellative semigroup? Let$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$into an commutative cancellative ... 0answers 66 views ### Hales' generalization of the stacked bases theorem (seeking a proof) In his paper Analogues of the stacked bases theorem, published in the proceedings of a 1976 conference, A.W. Hales claimed some interesting generalizations of the stacked bases theorem for abelian ... 1answer 362 views ### Abelian groups and their subgroups It is well known that every finite abelian group is a direct product of cyclic groups. So for every$n$every finite abelian group of exponent$n$is a direct product of cyclic groups of order at most ... 1answer 103 views ### Irreducible non-Abelian subgroup of$\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})$, that contains the subgroup of diagonal matrices. Does there exist any result ... 0answers 285 views ### A$\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$is an ... 0answers 244 views ### When is every element of a coend of abelian groups contained in one of the summands? Let$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$\bigoplus_{i \in I} D(i,i)$... 1answer 231 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 $$\mathit{... 0answers 139 views ### When is \{s_2-s_1,s_3-s_2,s_1-s_3\}\cap S non-empty for any s_1,s_2,s_3\in S? A subset S of an abelian group is a subgroup if and only if it is closed under taking differences; that is, the difference of any two elements of S is in S. Suppose, however, that we only know ... 0answers 107 views ### Invariant measure on coset space and integrable functions Let G be a locally compact abelian group, and H a closed subgroup. Let C_c(G) be the space of continuous, compactly supported complex valued functions on G. General theory of Haar measure ... 1answer 207 views ### Abelian torsion-free group with \mathbb{Z}_2\times\mathbb{Z} as automorphism group Let A be an abelian torsion-free group. If A is isomorphic with the group of rational numbers whose denominators are powers of, say, 2, then its automorphism group is isomorphic with \mathbb{Z}... 1answer 250 views ### Additive group of local rings Is there a theory or characterization for those finite p-groups that can be considered as the additive group of a finite local commutative ring with identity? 0answers 131 views ### Image of \rm{lim}^1 functor In category of abelian groups, we know that — values of \rm{lim}^1 on countable systems are precisely cotorsion groups — values of \rm{lim}^1 on systems of finitely generated groups are of the ... 0answers 89 views ### Name for a pair of lattices one of which having theta series with coefficients a subsequence of another lattice's theta series coefficients Is there a name for a pair of lattices which have the property given in the title (up to a change of variable)? The following example of a pair captures the property mentioned above:$$(i)\ 1 + 80q^3 ... 0answers 61 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$n$-... 1answer 306 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 ... 1answer 254 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 ... 2answers 393 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 ... 1answer 260 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 ... 1answer 135 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$...
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 \$...