Questions tagged [abelian-groups]
For questions about groups whose elements commute.
247
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Is $\mathbb Z$ prime in the class of abelian groups?
Let $B$, $C$, and $D$ be abelian groups such that $\mathbb Z\times B$ is isomorphic to $C\times D$.
Is there a group $E$ such that $C$ or $D$ is isomorphic to $\mathbb Z\times E$?
Reference: page 263 ...
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Alternating forms on abelian groups
Let $G$ and $H$ be abelian groups. By an alternating form, I mean a bilinear function $A\colon G\times G\to H$ such that $A(x,x)=0$ for all $x\in G$.
Question. If $A\colon G\times G\to H$ is an ...
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Adjoint to "strict twocategory of strict twofunctors"
Let C be the category of strict twofunctors, featuring the addition of a Grothendieck universe. Strict twocategories are categories enriched over the category of categories.
C has an internal hom ...
user531303
8
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1
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Bilinear forms on abelian $p$-groups: Images of weak metabolizers in the Frattini quotient
Suppose $G$ is a finite abelian $p$-group for $p$ an odd prime and $b : G \times G \to \mathbf{Q}/\mathbf{Z}$ is a non-degenerate symmetric bilinear form. A subgroup $H \leq G$ is a weak metabolizer ...
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Structure of a single automorphism of a finite abelian p-group
A finite abelian $p$-group $H$ is homogenous when it is the direct sum of cyclic groups of the same order $p^r$, i.e. $H \cong \big(\mathbb{Z}/p^{r}\mathbb{Z}\big)^{e}$. Every finite abelian $p$-group ...
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Can we test if an abelian group is finitely generated by taking tensor product?
If $A$ is a finitely generated abelian group, then we know that for all fields $k$, the tensor product $A\otimes_{\mathbb{Z}}k$ is finite dimensional as a $k$-vector space.
The converse is not true, ...
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"Universal" abelian p-groups
Let $p$ be a prime number. I am interested in the abelian groups $G$ with the following property:
(U) every finite abelian $p$-group $A$ admits a monomorphism $A\hookrightarrow G$.
In other words, ...
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Every abelian group can be embedded into a ring [closed]
Let $(G,0,+)$ be an abelian group. Does there always exist a ring with unity $(R,0,1,+,\cdot)$ and an injective homomorphism of groups $ \psi:(G,0,+)\rightarrow (R,0,+)$?
Is this hard to prove, or are ...
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2
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Maximal subgroups of finite abelian $2$-groups
Suppose $G$ is a finite abelian $2$-group, and $S$ is a subset of $G$, $\langle S\rangle=G$,$S^{-1}=S$,$e\notin S$. How to determine whether there exists a maximal subgroup $M$ of $G$, such that $S$ ...
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Small total variation distance between sums of random variables in finite Abelian group implies close to uniform?
Let $\mathbb{G} = \mathbb{Z}/p\mathbb{Z}$ (where $p$ is a prime). Let $X,Y,Z$ be independent random variables in $\mathbb G$.
For a small $\epsilon$ we have $\operatorname{dist}_{TV}(X+Y,Z+Y)<\...
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Computing the Abelian invariants of a subgroup of a f.g. Abelian group
We have a f.g. Abelian group $A$ given as a direct sum of $N$ cyclic subgroups $C_{k_j}=\langle x_j\rangle$, with $k_j\in \{2,\dots,\infty\}$, $1\leq j \leq N$, and the associated homomorphism $\phi:\...
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Isomorphic quotients of a countably infinitely-generated free abelian group
Let $F$ denote the free abelian group on countably infinite generators. I am trying to understand the relationship between normal subgroups $A$ and $B$ of $F$ with isomorphic quotients. So is there a ...
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Groups with prescribed Ulm invariants
In Kaplansky's book infinite abelian groups he provides (through some exercises) a complete classification of $p^{\infty}$-torsion countable abelian groups in terms of Ulm invariants. In other words ...
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203
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Descending chain in $\mathbb{Z}$ with certain confining property, but not strongly
Call a sequence $A_1 \supseteq A_2 \supseteq \cdots$ of subsets of $\mathbb{Z}$ confining if for all $i$ we have $A_i \supseteq A_{i+1}+A_{i+1}$. (Let us insist that the $A_i$ are symmetric and ...
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A question about freeness of a certain class of abelian groups
Lets call an abelian group $G$, to be semi-free (or SF) if every nonzero subgroup of $G$ is isomorphic to $\mathbb{Z}\times H$ for some abelian group $H$.
Is every semi-free group, a free group? If ...
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