# Questions tagged [abelian-groups]

For questions about groups whose elements commute.

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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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### 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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### Can a non-free Whitehead group embed as a discrete subgroup of a normed space?

Every countable discrete subgroup of a normed space is isomorphic to the direct sum of the group of integers. I wonder whether it is possible to push this beyond such direct-sum (free abelian) groups ...
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### Almost free group without the Specker group as a subgroup

An Abelian group is almost free whenever every countable subgroup is free Abelian. Famously, the Specker group $\mathbf Z^{\mathbf N}$ is almost free. What are examples of almost free groups that are ...
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Let $G$ be a reduced abelian $p$-group. We set $G_0=G$. Let $\alpha$ be an ordinal. Inductively, if $\alpha=\beta+1$ is a successor ordinal, we define $$G_{\alpha}=pG_{\beta}.$$ If $\alpha$ is a limit ...
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### Decomposition of finite abelian groups of even order if there is an involution

Let $G$ be a finite abelian group and $\sigma :G\rightarrow G$ an automorphism of order two ($\sigma\circ \sigma =id_G$). Denote by $F$ and $A$ the subgroups of fixed and anti-fixed points of $\sigma$ ...
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### 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 ...
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### 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 ...
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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
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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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### 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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### 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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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 ...