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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.

An abelian or commutative group is a group $(G,*)$ in which all elements commute: $$\forall a,b\in G\,\,, a*b=b*a\,.$$ Usually the product is denoted by $+$ in an abelian group, and the identity of the group by $0$. Abelian groups are also known as modules over the ring $\mathbb{Z}$ of integers.

Examples include the integers $\mathbb{Z}$ under addition, as well as the rationals $\mathbb{Q}$ under addition. In fact, every cyclic group is an abelian group. Non-examples include $S_3$, the symmetry group on three elements, as well as $\mathrm{SO}(3)$, the rotations in three dimensions.

The fundamental theorem of abelian groups says that all finite abelian groups are direct products of cyclic groups, themselves abelian.

3991 questions
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Proof about existence of subgroups in finite abelian group

I am trying to understand the proof of the following theorem. Theorem: If $G$ is a finite abelian group and $d$ is a divisor of $|G|$, then $G$ contains a subgroup of order $d$. Proof: Let $d$ be any divisor of $|G|$, and let $p$ be a prime divisor…
YYF
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Showing that any group of order 286331153 is abelian

This is the third part of a set of problems, of which I have solved 2. I have shown that if $p$ is prime, the group $Aut(\mathbb Z_p)$ is of order $p-1$. I have shown that $Aut(\mathbb Z_{17})$, $Aut(\mathbb Z_{257})$, $Aut(\mathbb Z_{65537})$ are…
Alec
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A Counter example for direct summand

All my attempts proving the following claim have been useless and it seems to be wrong, but can not find any counter example(s) for it. :) "If U and N be two direct summands of an abelian group G such that N+U=G, then the intersection of N and U is…
Nancy Rutkowskie
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Size of generating sets for subgroups of finitely generated abelian groups.

If $A$ is a f.g. abelian group, let $\mu(A)$ be the minimal number of elements needed to generate $A$. If we know $\mu(A)$, what is $M(A)=\sup\limits_{B\subseteq A}\mu(B)$, with the supremum taken over all subgroups $B\subseteq A$?
Mariano Suárez-Álvarez
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setting abelian group in non-abelian group

Is it right to say for every (finite) abelian group $H$ there is non-abelian group $G$ such that $Z(G)=H$, where $Z(G)$ is the center of $G$?
user148528
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Torsion-free abelian group of rank 1

I find it hard to understand a part of a proof on torsion-free abelian groups of rank 1. Let $A$ and $B$ be torsion-free groups of rank 1 and of the same type. Let $a'$ and $b'$ be arbitrary non zero elements from $A$ and $B$, respectively. Then the…
stacy
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Show that the normal subgroup is cyclic

Let $G=\mathbb Z\times\mathbb Z$. Consider $H\leq G$ generated by $(-5,1)$ and $(1,-5)$. Show that $\frac{G}H$ is cyclic. This is what I have so far but I'm not sure if I'm right either. Let $a,b,\in\mathbb Z$. So $a(-5,1)+b(1,-5)=(b-5a,a-5b).$…
Haikal Yeo
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Does $0 \to H \to G \to G/H\to 0$ split for $H$ with special property?

Let $G$ be an abelian group and $H \subseteq G$ a subgroup with the following property: $$ H = \{ g \in G \mid \exists n\in \mathbb N^{+} \: ng \in H \}.$$ Does the short exact sequence $0 \to H \to G \to G/H\to 0$ split? Or in other words: Do we…
principal-ideal-domain
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homomorphisms of abelian groups

Describe: 1) Hom(Q/Z->Q) 2) Hom(Q->Q/Z) Q rational numbers Z integers My thoughts 1) Q/Z i can describe as a/b, a,b coprime and smaller than 1. So I thought Hom(Q/Z->Q) I can describe as [f:a+N->b, N subgroup of Q/Z, a element of Q, b any element of…
user60419
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Torsion subgroup of $\mathbb{C}^\times$

I need to find the torsion subgroup of the multiplicative abelian group $\mathbb{C}^\times$. This is from a homework assignment sheet, and I'm not sure what the notation $\mathbb{C}^\times$ stands for. I'm assuming it's the group of units. Every…
Lammey
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Does there exist a complement of a subgroup in a abelian group.

Let $G$ be an abelian group and $H$ subgroup of $G$. Suppose that: (i) $H$ has a complement in $G$. (ii) $K$ is a subgroup of $G$ and K is isomorphic to $H$ Is there a complement of $K$ in $G$? If yes, what is the relation of complements of $H$ and…
Luyen Le
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order of an element of a group and its coset

I find it hard to understand a part of the proof of the existence of any basic subgroup in every abelian torsion group.I'm going to write you the information I think useful. Let $G$ an abelian torsion group. Let $B=\langle X\rangle$ where $X$…
stacy
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Is this claim true about extention of $Z$ by $Z$?

Kindly ask this question: Can we say that $\mathbb{Z}$ × $\mathbb{Z}$ is an extention of $\mathbb{Z}$ by $\mathbb{Z}$?
Samir
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Is this deduction valid for a abelian group G and dG?

We know that, any abelian group G can be written as a direct sum of dG and a reduced subgroup R (where dG is the subgroup generated by all divisible subgroups of G). Is it true that R is isomorphic to G/dG?
Basil R
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Showing a quotient group $G/\langle x_0\rangle$ to be a direct sum of cyclic groups

There is an infinite p-primary group which is not isomorphic to $\mathbf{Z}_{p^∞}$ : $G=\langle x_0,x_1,\ldots\mid px_0, x_0-p^nx_n\rangle$ for all $n\geq 1$. Can I ask any hints, for showing that $G/\langle x_0\rangle$ is a direct sum of cyclic…
Nancy Rutkowskie
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