Should be used with the (group-theory) tag. A group $(G,*)$ is said to be abelian if $a*b=b*a$ for all $a,b\in G.$

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42
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The direct sum $\oplus$ versus the cartesian product $\times$

In the case of abelian groups, I have been treating these two set operations as more or less indistinguishable. In early mathematics courses, one normally defines $A^n := A\times A\times\ldots\times ...
26
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2answers
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Structure Theorem for abelian torsion groups that are not finitely generated

I know about the structure theorem for finitely generated abelian groups. I'm wondering whether there exists a similar structure theorem for abelian groups that are not finitely generated. In ...
26
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4answers
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Prove that if $(ab)^i = a^ib^i \forall a,b\in G$ for three consecutive integers $i$ then G is abelian

I've been working on this problem listed in Herstein's Topics in Algebra (Chapter 2.3, problem 4): If $G$ is a group such that $(ab)^i = a^ib^i$ for three consecutive integers $i$ for all $a, b\in ...
24
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2answers
521 views

New twist on a Putnam problem

A recent Putnam problem: Let $f$ be a real-valued function on the plane such that for every square $ABCD$ in the plane, $f(A)+f(B)+f(C)+f(D)=0$. Does it follow that $f$ is identically zero? The ...
22
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2answers
951 views

In a group we have $abc=cba$. Is it abelian?

Let $G$ be a group such that for any $a,b,c\ne1$: $$abc=cba$$ Is $G$ abelian?
22
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1answer
643 views

Recovering a finite group's structure from the order of its elements.

Suppose you know the following two things about a group $G$ with $n$ elements: the order of each of the $n$ elements in $G$; $G$ is uniquely determined by the orders in (1). Question: How ...
18
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2answers
245 views

Are these exactly the abelian groups?

I'm thinking about the following condition on a group $G$. $$(\forall A\subseteq G)(\forall g\in G)(\exists h\in G)\ Ag=hA.$$ Obviously every abelian group $G$ satisfies this condition. Are ...
17
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3answers
567 views

How “abelian” can a non-abelian group be?

Something I have been wondering: in general, is there a bound for how many elements in a finite non-abelian group $G$ can commute with every other element? Equivalently, is there is a bound for the ...
17
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2answers
333 views

If the tensor power $M^{\otimes n} = 0$, is it possible that $M^{\otimes n-1}$ is nonzero?

Let $M$ be a module over a commutative ring $R$. It is possible that $M \otimes M = 0$ if $M$ is nonzero, for example when $R = \mathbb{Z}$ and $M = \mathbb{Q}/ \mathbb{Z}$. What about when higher ...
14
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1answer
510 views

Are $(\mathbb{R},+)$ and $(\mathbb{C},+)$ isomorphic as additive groups?

Are $(\mathbb{R},+)$ and $(\mathbb{C},+)$ isomorphic as additive groups? I know that there is a bijection between $\mathbb{R}$ and $\mathbb{C}$, and this question asks whether they are isomorphic as ...
14
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4answers
883 views

How many non-isomorphic abelian groups of order $\kappa$ are there for $\kappa$ infinite?

Let $\kappa$ be an infinite cardinal. How many non-isomorphic abelian groups of order (cardinality) $\kappa$ are there? For finite $\kappa,$ we can use the classification theorem and obtain the ...
13
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2answers
971 views

Why is every abelian group the direct limit of its finitely generated subgroups?

I'm taking classes in homological algebra now, and the book (together with the lecturer) seem to assume more category theory than I already know. A "fact" that is used freely in the book ...
13
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2answers
389 views

Status of the classification of non-finitely generated abelian groups.

From the Wikipedia on abelian groups: By contrast, classification of general infinitely-generated abelian groups is far from complete. How far are we from a classification exactly? It seems ...
13
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1answer
1k views

Is it true that $\mathbb{R}$ and $\mathbb{R}^2$ are isomorphic as abelian groups?

I think the answer is yes. Sketch of the proof Consider $\mathbb{R}$ as a vector space over $\mathbb{Q}$. Let $\{e_\lambda:\lambda\in\Lambda\}\subset\mathbb{R}$ be its Hamel basis. Then ...
12
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1answer
190 views

Problem on abelian group

Let $G$ be an abelian group, and $\Phi:G\to \mathbb{R}$ is a function with the following property: $$\forall a,b\in G,~~ |\Phi(a+b)-\Phi(a)-\Phi(b)|<c$$ The problem asks to prove the existence of ...
12
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2answers
235 views

$G$ abelian group with $\vert G\vert =mn$ and $\gcd(m,n)=1$.

I hope this is not a duplicate. First of all, in what follows I'm not allowed (unfortunately) to use the structure theorem for abelian groups. I'm asked to prove the following: Let $G$ be an ...
11
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2answers
834 views

Find an abelian infinite group such that every proper subgroup is finite

I found this question in Arhangel'skii and Tkachenko's book Topological Groups and Related Structures. The first chapter of the book is devoted to algebraic preliminaries. The question actually ...
11
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3answers
287 views

If $|\lbrace g \in G: \pi (g)=g^{-1} \rbrace|>\frac{3|G|}{4}$, then $G$ is an abelian group.

Assume that $\pi$ is an automorphism of a finite group $G$. Let $S$ denote the set $\lbrace g \in G: \pi (g)=g^{-1} \rbrace$. Show that if $|S|>\frac{3|G|}{4}$, then $G$ is an abelian group. ...
11
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3answers
268 views

Non-Abelian groups and subgroups

Need to find example for non-abelian group $G$ for which $A \subset G,\;A=\{g\in G \mid g^{-1}=g\}$ is not a subgroup of $G$. Can you please help me find such a $G$?
11
votes
1answer
231 views

Is there a topological group that is connected but not path-connected?

Is there a $\big($T$_0$$\hspace{-0.02 in}\big)$ topological group that is connected but not path-connected? If yes: $\quad$ Can it be complete? $\:$ (with respect to the two-sided uniform structure) ...
10
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2answers
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Group of positive rationals under multiplication not isomorphic to group of rationals

A question that may sound very trivial, apologies beforehand. I am wondering why $( \mathbb{Q}_{>0} , \times )$ is not isomorphic to $( \mathbb{Q} , + )$. I can see for the case when $( \mathbb{Q} ...
10
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2answers
170 views

The existence of a group automorphism with some properties implies commutativity.

Let $G $ be a finite group, $T$ be an automorphisom of $ G $ st $ Tx = x \iff x=e $. Suppose further that $ T^2 =I $. Prove that $ G $ is abelian. I was thinking if I show $ T aba^{-1} b^ ...
10
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2answers
231 views

Applications of the fact that a group is never the union of two of its proper subgroups

It is well-known that a group cannot be written as the union of two its proper subgroups. Has anybody come across some consequences from this fact? The small one I know is that if H is a proper ...
9
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5answers
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Are cyclic groups always abelian?

If a group $C$ is cyclic, is it also abelian (commutative)? If so, is it possible to give an “easy” explanation of why this is? Thanks in advance!
9
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4answers
398 views

Coproducts in $\text{Ab}$

I am currently trying to understand why finite products and coproducts in the category $\text{Ab}$ coincide. In fact, I'm not even sure I can show it. My question is the following: Is there an ...
9
votes
3answers
316 views

Additive group of rationals has no minimal generating set

In a comment to Arturo Magidin's answer to this question, Jack Schmidt says that the additive group of the rationals has no minimal generating set. Why does $(\mathbb{Q},+)$ have no minimal ...
9
votes
1answer
120 views

Prove $G$ is abelian if $f(f(x)) = x$?

Let $G$ be a finite group and $f$ an automorphism such that $f(f(x)) = x$, and $f(x) = x$ if and only if $x=e$. Prove that $G$ is abelian and $f(x) = x^{-1}$. My attempt: ...
9
votes
1answer
175 views

Finding the order of the automorphism group of the abelian group of order 8.

So I am given an abelian group of order $8$ such that for all non-identity elements $x^2 = e$ (all elements have order two). So I know the answer is gonna be $168$, but I gotta prove this. So far I ...
9
votes
1answer
401 views

Does the splitting lemma hold without the axiom of choice?

In part of the proof of the splitting lemma (a left-split short exact sequence of abelian groups is right-split) it seems necessary to invoke the axiom of choice. That is, if $0\to A\overset{f}{\to} ...
9
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1answer
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If $G/Z(G)$ is cyclic, then $G$ is abelian

Continuing my work through Dummit & Foote's "Abstract Algebra", 3.1.36 asks the following (which is exactly the same as exercise 5 in this related MSE answer): Prove that if $G/Z(G)$ is ...
8
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5answers
661 views

Is $\mathbb{Z}^2$ cyclic?

Is $\mathbb{Z}^2$ cyclic? What does it mean for a group to be cyclic? Is it just that it has one generator? Thanks
8
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3answers
379 views

Finding an explicit isomorphism between $\mathbb{Z}^{\times}_n$ and $\mathbb{Z}^{\times}_{2n}$

For an odd integer $n$, find an explicit isomorphism between $\mathbb{Z}^{\times}_n$ and $\mathbb{Z}^{\times}_{2n}$. How do I do this? I don't really know where to start. I can easily find bijections ...
8
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1answer
176 views

For which $n$, $G$ is abelian?

My question is: For Which natural numbers $n$, a finite group $G$ of order $n$ is an abelian group? Obviouslyو for $n≤4$ and when $n$ is a prime number, we have $G$ is abelian. Can we consider ...
8
votes
2answers
340 views

Does every ring with unity arise as an endomorphism ring?

I don't believe that every ring with a $1$ is the endomorphism ring of an abelian group but I currently don't see how to produce a counterexample.
8
votes
3answers
173 views

Tensor-commutative abelian groups

Say that an abelian group $A$ is tensor-commutative if the equality $x\otimes y=y\otimes x$ holds in $A\otimes_{\mathbb Z}A$ for all $x,y$ in $A$. The first question is somewhat vague: Question 1. ...
8
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3answers
139 views

Subgroups of $\mathbb{Z}_2 \times \mathbb{Z}_{12}$ of order $6$

what are the subgroups of $\mathbb{Z}_2 \times \mathbb{Z}_{12}$ of order $6$? I know that there are three such subgroups, and two subgroups are clear to me, namely the subgroup isomorphic to ...
8
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1answer
629 views

Finite abelian groups - direct sum of cyclic subgroup

Let $G$ be a finite abelian $p$-group. It is quite elementary to see that if $g \in G$ is an element of maximal order (and thus its span is a cyclic subgroup of $G$ of maximal order) then $G$ can be ...
8
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2answers
556 views

Can the semidirect product of two groups be abelian group?

while I was working through the examples of semidirect products of Dummit and Foote, I thought that it's possible to show that any semdirect product of two groups can't be abelian if the this ...
8
votes
1answer
74 views

Equality of rank for homology and cohomology groups via the universal coefficient theorem

I'm having trouble understanding a passage from the proof of Corollary 3.37 in Hatcher's Algebraic Topology, namely the fact that the universal coefficient theorem implies $$ ...
8
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1answer
94 views

Cyclic subgroups of $GL_2(\mathbb{Z}/p\mathbb{Z})$?

What are the cyclic subgroups of $GL_2(\mathbb{Z}/p\mathbb{Z})$, the general linear group over the finite field of order $p$, where $p$ is prime? Obviously, each cyclic subgroup is generated by some ...
8
votes
3answers
944 views

A nonsplit short exact sequence of abelian groups with $B \cong A \oplus C$

A homework problem asked to find a short exact sequence of abelian groups $$0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$$ such that $$B \cong A \oplus C$$ although the sequence does ...
8
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1answer
121 views

on finite abelian groups

Let $G$ be a finite abelian group and let $M(G)$ be the set of all elements of $G$ that fix with any automorphism of $G$. Then prove $$M(G)=\langle1\rangle \text{ or } Z_{2}$$ Attempt: We know that ...
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2answers
164 views

Structure of the group of arithmetic functions

This question was originally posted in Elements of finite order in the group of arithmetic functions under Dirichlet convolution. and it goes as follows: Let G be the group consisting of all ...
7
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3answers
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Prove that if $g^2=e$ for all g in G then G is Abelian.

This question is from group theory in Abstract Algebra and no matter how many times my lecturer teaches it for some reason I can't seem to crack it. (please note that $e$ in the question is the ...
7
votes
4answers
171 views

Let $G$ be a finite group which has a total of no more than five subgroups. Prove that $G$ is abelian.

Let $G$ be a finite group which has a total of no more than five subgroups. Prove that $G$ is abelian. I can prove that if $\left|G\right|\leq5 $ then $G$ is abelian. Is it equivalent to this ...
7
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3answers
233 views

The Hopfian property for groups

Let $G$ be a group, which for my purposes would be abelian. To say that $G$ has the Hopf property is to say that every epimorphism of $G$ is an automorphism. Does anyone happen to recall the context ...
7
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3answers
106 views

For any $n$, $G^\dot{n}$ is a subgroup of $G$. Is $G$ abelian?

Let $G$ be a group such that for each $n$: $$G^\dot{n}=\{a^n|a\in G\}$$ is a subgroup of $G$. Is $G$ abelian?
7
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2answers
534 views

Can the tensor product of two non-free abelian groups be non-zero free?

It's pretty easy to construct an (R-S) bi-module M and a left S-module N such that neither M_S nor N is a projective S-module, but the tensor product is a non-zero projective R-module. However, ...
7
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2answers
246 views

Facts about Abelian Groups and group order.

I look for some theorems which tell us about the relation between the property of being abelian for groups and the order of the group. I think these theorems are provided in a second course of group ...
7
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2answers
364 views

Abelian group admitting a surjective homomorphism onto an infinite cyclic group

I am working on the following problem: Let $G$ an Abelian group and $f: G \to \Bbb Z$ a surjective homomorphism. Prove that $G \cong \ker(f) \times \Bbb Z$ By means of the First Isomorphism ...