# 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.

3,535 questions
Filter by
Sorted by
Tagged with
33 views

### When all commutant (centralizer) subgroups are abelian

I have seen the following problem in chapter 9 of Abstract Algebra by Dan Saracino: Let $G$ be a group and for $a,b \in G$ let $a\ R\ b$ mean that $ab=ba$. Must $R$ be an equivalence relation on $G$? ...
• 1,035
46 views

• 364
29 views

• 35
18 views

• 23
76 views

42 views

• 1,349
33 views

### The cartesian product of an abelian and non abelian group [closed]

If I take the Cartesian product of two groups, with one being abelian and the other being non abelian, Is the product always abelian, always non abelian, or can it be either?
1 vote
48 views

### The equational and quasi-equational theory of commutative groups in the signature $*$.

I know that the equational theory of groups in the signature $*$ is axiomatized by the associative law, so does that mean the equational theory of commutative groups in the signature $*$ is ...
• 13.2k
1 vote
59 views

### Torsion free group is Abelian if $(gh)^r=g^rh^r$ [closed]

Let $G$ be a torsion free group where for some fixed $r\ge2$ we have the identity $(xy)^r=x^ry^r$ for all $x,y\in G$. Does this imply $G$ is Abelian?
• 187
28 views

873 views

### Cannot become ring because distribution law does not hold

Commutative ring with unit is defined as $(R,+,\times)$, where $(R,+)$ is abelian group and $(R,\times)$ is commutative multiplicative monoid with $1$ and $+$ and $\times$ satisfies distributive law. ...
• 574
1 vote
73 views

### In group-theory, are the elements in a set other sets or are they precise numbers?

I am starting out with group theory for my computer science degree, it's part of the basic maths subject, it is covered in about 1.5 pages and then moves on with topology. From what I understood you ...
I needed to find the number of Abelian groups of order $n = 2^3 \cdot 3^4 \cdot 5^6 \cdot 7^{11}.$ And then to find in how many of those groups the maximum order of an element is $\leq$ $n \div 12$. I ...
Let $G$ be a finitely generated abelian group. Prove that the quotient group $G/2G$ is finite. I tried two approaches but did not succeed. Structure Theorem: \$G\cong\mathbb{Z}^r\times\prod\mathbb{Z}...