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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.

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Does $G\oplus G$=$H\oplus H$ imply $G$=$H$ for divisible abelian groups?

Suppose $G$ and $H$ are two divisible abelian groups. Futhermore we have $G\oplus G$=$H\oplus H$. Is $G$ isomorphic to $H$?
Basil R
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Expressing an abelian group as a sum of cyclic groups

Find an isomorphic direct product of cyclic groups, where $V$ is an abelian group generated by $x,y,z$ and subject to relations: $$3x + 2y + 8z = 0,\qquad 2x + 4z = 0.$$ The answer is $C_{4} \oplus \mathbb{Z}$ But I don't know how to get the…
user8603
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When is $F(A)$ an abelian group?

Let $A$ be a nonempty finite set. Denote by $F(A)$ the set of all bijections $f : A \to A$. When is $F(A)$ an abelian group?
Peter Liu
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About Classification of Finitely Generated Abelian Groups

I am studying Finitely Generated Abelian Groups. Now I find a material of Wolf Holzmann abelian.pdf I have a question in this material: Can I replace all notation $\oplus$ by $\times$?. More precisely, Can I replace $K\cong d_1 \mathbb{Z}\oplus…
Muniain
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Are all groups of order $2\cdot p$ abelian?

Are all groups of order $2\cdot p$ abelian? I would like to prove that Dihedral group $D(p)$ is the only non-abelian group of that order. $p$ is any prime $>2$. Thanks for advice.
Tim Beno
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Prove that if a and b are elements of an Abelian group G, with o(a) = m and o(b) = n, then (ab)^{mn}

(a) Prove that if a and b are elements of an Abelian group G, with o(a) = m and o(b) = n,then (ab)^{mn} = e. Indicate where you use the condition that G is Abelian. (b) With G, a, and b as in part (a), prove that o(ab) divides o(a)o(b). (c) Give an…
sh.alzoubi
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Quotient group of free abelian group

Let $A$ be a free abelian group, i.e. $A=\bigoplus_\alpha \mathbb Z$. Also let $B$ be a subgroup of $A$. Prove that $A/B\cong\mathbb Z$ implies $A=B\oplus \mathbb Z$. p.s. Actually this appears in Hatcher's Algebraic Topology where he says…
Y.H. Chan
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Are all complete groups abelian?

Hi: Let $G$ be a group and $G'$ it's commutator subgroup. Then $G > G' > 1$ is a series of normal subgroups of $G$. Suppose $G$ is complete. Then, if I'm not wrong, $Aut(G)$ is the stabilizer of that series. But then, by a theorem, $Aut(G)$ is…
stf92
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Maximal subgroups of Z

In the ring of integers, the only maximal ideals are those generated by the prime elements. Is the same true for the group of integers? Are the only maximal subgroups of integers the ones generated by the primes?
Reem C
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Exercise 6.79 from Rotman's Advanced Modern Algebra

If $G$ is a nonzero abelian group show that $$\operatorname{Hom}_{\Bbb Z}(G,\frac{\Bbb Q}{\Bbb Z}) \neq \{0\}.$$
erfan
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$A/A^{p}\cong A_{p}$ for finite abelian (additive) gp. and prime $p$.

Let $A$ be a finite abelian (additive) gp. and $p$ be a prime. I want to show $A/A^{p}\cong A_{p}$ where $A^{p}:=\left\{pa:a\in A\right\}$ and $A_{p}:=\left\{a\in A:pa=0\right\}$.(I want to show $A/A^{p}\cong A_{p}$ not $A/A_{p}\cong A^{p}$) To show…
jawlang
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How do I check this simple set is an Abelian group?

The n-gon in question is a 3-gon. It is an equilateral triangle to be exact. This is a Dihedral group of order 6 (3 reflections and 3 rotations) I have plotted the Cayley's table. The set of elements in $$D_3$$ is…
Mathematicing
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Classifying some abelian groups of order $2^5\times 3^5$

I'm requested to classify the abelian groups $A$ of order $2^5 \times 3^5 $ where : $| A/A^4 | = 2^4 $ $ |A/A^3 | = 3^4 $ I need to write down the canonical form of each group . My question is, what does it mean $| A/A^4 |$ ? I understand that…
JAN
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Abelian group, inverse element

What would be the inverse element for this abelian group $[1,2,3,4,...,p-1]$ , in which p is a prime number, with this operation $(a*b)mod.p$? For all $a$ and $b$ of the set. I know the inverse element has to be a multiplication of $p$ plus $1$,…
Arthur
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Rational rank of l-group

Is it possible to have a finitely generated lattice - ordered group has rational rank bigger than the number of generators of the group?
Rajesh
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