Questions tagged [abelian-groups]

For questions about abelian groups, including the basic theory of abelian groups as a topic in elementary group theory as well as more advanced topics (classification, structure theory, theory of $\mathbb{Z}$-modules as related to modules over other rings, homological algebra of abelian groups, etc.). Consider also using the tag (group-theory) or (modules) depending on the perspective of your question.

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When is the abelian groups homomorphism $g \mapsto 2g$ injective?

This is possibly an easy question. Let $(G,+)$ be a countable abelian group. Under what circumstances is the homomorphism $\phi: G \mapsto 2G:=\{g+g: g\in G\}$, defined by $\phi(g)=g+g$ injective? I ...
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Uniqueness of module structure

Question: Let $E$ be a torsion abelian group. Prove that $E$ has exactly one $\hat{\mathbb{Z}}$-module structure, and that the scalar multiplication $\hat{\mathbb{Z}} \times E \rightarrow E$ defining ...
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If $(ab)^2=b^2 a^2 , \forall a , b\in G$ , then $G$ is necessarily abelian?

Let $G$ be a group such that $(ab)^2=b^2a^2 \forall a,b\in G$ then $G$ is abelian? I tried to find a counterexample, but nothing came up. Hence I tried to prove it. We have $(aba^-)^2=ab^2a^-$ and ...
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Baer's definition of $S$-characteristic subgroup

I am reading a part of Baer's paper Groups without proper isomorphic quotients, Bull. AMS,50 (1944) 267-278. In that paper, he also discusses groups without proper isomorphic subgroups, which he calls ...
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Pre-Sheaf on locally compact abelian group

Let $G$ be a locally compact abelian group. We can define a pre-sheaf $\mathcal{F}$ on G by $$\mathcal{F}(U) := \widehat{\langle U\rangle}$$ for any open set $U$, where $\widehat{\langle U\rangle}$ ...
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Topological tensor product

Let $A$ and $B$ be topological abelian groups. Consider the following universal property for a pair $(P, \pi)$, where $P$ is a topological abelian group and $\pi: A \times B \to P$ is a jointly ...
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KL is an abelian extension implies K and L are abelian extensions

Prove of give a counterexample: If $KL$ is abelian then $K$ and $L$ are abelian. I have proven the converse, that is if $K$ and $L$ are abelian then $KL$ is abelian. I have a feeling the above doens'...
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metabelian groups can be represented by matrices, do we know the exact representation?

We know that every finitely generated metabelian group can be represented by matrices (e.g. see https://link.springer.com/article/10.1007/BF02219822 ). I am particularly interested in how finitely ...
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Prove that $\sum_{i=1}^{k} \lambda_i f(g_i x) \geq 0$ holds for all $x \in G$, then $\sum_{i=1}^{k} \lambda_i \geq 0.$

Problem statement: Let $f(x) \geq 0$ be a nonzero, bounded, real function on an Abelian group $G$, $g_1, \ldots, g_k$ are given elements of $G$, and $\lambda_1, \ldots, \lambda_k$ are real numbers. ...
1 vote
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Let $G$ be a finite abelian group of order n. How many distinct group homomorphisms are there from $G$ to $R/Z$?

First, is this question well-defined? That is, the homomorphisms are determined by the order n. Since $R/Z$ is the multiplicative unit circle, all finite subgroups are cyclic, so the image of the ...
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Is every element of a finite abelian group contained in a cyclic factor?

Suppose $A$ is a non-cyclic finite abelian group and $a\in A$. Can we always find $A_1,A_2\leq A$ such that $a\in A_1$ and $A=A_1\oplus A_2$? Equivalently, is it true that given a finite abelian $A$ ...
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Proof of the universal property of free abelian groups

Let $S$ be a set. The group with presentation $(S, R)$ , where $R = \{\ [s , t] \mid s,t\in S\ \}$ is called the free abelian group on $S$ -- denote it by $A (S)$ . Prove that $A (S)$ has the ...
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