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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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Isomorphisms in finite abelian groups 1

True of false? If G and H are two groups with the same order and both are abelian, then they are isomorphic.
ronil
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Finding the order of a subgroup of a finite abelian group as a direct sum of cyclic groups

Let $G$ be an abelian group of order $p^m$. Suppose that $G$ is a direct sum of $t$ cyclic groups. Show that the subgroup $H$ of $G$ containing $0$ and order $p$ elements has order $p^t$. By the structure theorem of finite abelian groups, we…
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A finite abelian group question

In homework of the book 'Introduction to algebra and finite fields', I have a question about finite group. There is the question: Suppose $G$ is a finite Abelian group. For any $a,b\in G$, there exists $c\in G$ such that $o(c)=[o(a),o(b)]$. o(c) is…
zkp_learner
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A torsion-free group $G$ is imbedded in a Vector space over $\mathbb Q$ of dimention $n$?

We know that any torsion-free group can be imbedded in a vector space over $\mathbb Q$. Now the question is: if there is a maximal independent subset with $n$ elements of our torsion-free group $G$,$X$, then the group can be imbbeded in a vector…
Mikasa
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Quotient by a torsion group

Let $A$ be a finitely generated abelian group of rank $r$. The rank of the abelian group $A$ is the number of copies of $\mathbb Z$. Let $T$ be the torsion subgroup of $A$. Show that $\frac{A}{T(A)}\cong\mathbb Z^r$. I don't know if it helps but…
Haikal Yeo
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Let $G$ be a group in which $a^2=e$ for all elements of $a$ of $G$. Show that $G$ is Abelian.

Let $G$ be a group in which $a^2=e$ for all elements of $a$ of $G$. Show that $G$ is Abelian. I need help on this problem. Appreciated!
user67253
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Subgroups of torsion abelian groups

I want to know any subgroups of the infinite direct sum of the abelian group $\Bbb{Z}_{p^n}$ where $p$ is a prime number.
Najmeh
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Finite Abelian group and subgroup

Suppose $G$ is a finite abelian group and $H$ be a proper subgroup. Let $a$ be an element in $G$ not in $H$. Does there always exists an $m>0$ s.t. $a^m \in H$? If it is there what is the proof?
roydiptajit
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Prove a finitely generated abelian group in which every element is of finite order is finite.

Let $A$ and $B$ are free $\mathbb{Z}$-module and has the same rank $n$. If rank $A/B＝0$, then $|A/B|$ is finite ? From rank $A/B＝0$, I could deduce that orders of every element $A/B$ is finite. But from this, I think “$|A/B|$ is finite” does not…
user695664
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Is there an abelian linearly ordered group of the multiplicative convention for complex number excluding $\{0\}$?

Evidently, $\{0\}$ has to be excluded since it has no inverse. My question is reduced to: Is there any total order on complex numbers w/o $0$? From what I sense (but not 100% sure), the lexicographical order (also known as lexical order) is a total…
Jan Tamariani
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Prove if $G$ is abelian, $g\in G$ is of maximal order, and $h \in G$ has finite order, then $|h|$ divides $|g|$.

I've been playing around with the problem for a while now but haven't managed to make any progress. I would really appreciate a nudge in the right direction -- but please no full solutions. Attempt: Assume $|h|$ is finite but does not divide $|g|$.…
Andrew Tawfeek
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Infinite abelian group splits over pure subgroup

Given an abelian group $G$ with pure subgroup $A$ with $[G:A]<\infty$. Show that we can find a subgroup $B$ such that $G$ is the direct sum of $A$ and $B$. I can see how to do it for a divisible subgroup (by showing that the subgroup is injective),…
almagest
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Why $Hom(\mathbb{Q},G)=0$?

Let $F$ be the free abelian group generated by $X$, where $X=\{a,b_{1},b_{2},...,b_{n}...\}$ and let $K=\langle Y \rangle$. Here $Y=\{2a, a-2^{n}b_{n},n\ge 1\}$. Define $G=F/K$. Now I am supposed to prove: $$0\rightarrow \langle a \rangle…
Bombyx mori
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Abelian groups of odd order

If G is a finite abelian group of odd order prove that the product of all elements in G is equal to the identity element of G.Well , I can prove this with an example but is there any general proof for this?
Shashank Dwivedi
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How to prove that a group is abelian if $a^2=1$ for all $a \in A$.

Suppose that $G$ is a group with the property that $g^{2}= 1$ for all $g \in G$.Prove that $G$ is a commutactive group. Abelian group $ab = ba$. I think like this $g*g^{-1} = 1$ after that i get stuck any hints?
user3704516
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