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

learn more… | top users | synonyms

48
votes
3answers
5k views

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 ...
29
votes
4answers
4k views

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 ...
27
votes
2answers
2k views

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
votes
2answers
1k 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?
26
votes
2answers
582 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
votes
1answer
695 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 ...
19
votes
1answer
817 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 ...
18
votes
2answers
267 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
votes
2answers
394 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 ...
17
votes
3answers
722 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
votes
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 ...
16
votes
2answers
1k 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 ...
14
votes
4answers
1k 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
votes
2answers
1k 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 ...
13
votes
2answers
574 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 ...
12
votes
1answer
213 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
votes
1answer
305 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) ...
12
votes
2answers
270 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
votes
3answers
6k views

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 ...
11
votes
2answers
2k views

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} ...
11
votes
3answers
409 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
votes
3answers
302 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
7k views

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 ...
11
votes
2answers
285 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 ...
11
votes
3answers
1k 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 ...
11
votes
3answers
3k views

Let $C$ be the commutator subgroup of $G$. Prove that $G/C$ is abelian

Trying to get my head around the commutator subgroup. This is an excercise from Artin's Algebra: Let $C$ be the commutator subgroup of $G$. Prove that $G/C$ is abelian. Here is what I've done: Let ...
10
votes
5answers
4k views

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!
10
votes
4answers
480 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 ...
10
votes
2answers
183 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
votes
1answer
139 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: ...
10
votes
1answer
214 views

Writing $G/A\times G/B$ explicitly as union of orbits

Let $G$ be a finite abelian group, and let $A$ and $B$ be subgroups. I'm interested in $G/A\times G/B$ with its natural $G$-set structure. In $G/A\times G/B$, the stabilizer of any element is $A\cap ...
9
votes
4answers
2k views

Show that $({\mathbb{Q}},+)$ is not finitely generated using the Fundamental Theorem of Finitely Generated Abelian Groups.

Can anyone please help me out on how to use the fundamental theorem of finitely generated abelian groups to prove that $({\mathbb{Q}},+)$ is not finitely generated?
9
votes
4answers
562 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
2answers
143 views

Is $\Bbb Q/\Bbb Z$ artinian as a $\Bbb Z$-module?

I'm confused. Is $\Bbb Q/\Bbb Z$ artinian as a $\Bbb Z$-module? We know that $\Bbb Z_{p^{\infty}} \subset \Bbb Q/\Bbb Z$ is artinian. The following argument is true or not ? $\mathbb Q / ...
9
votes
1answer
189 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 ...
9
votes
2answers
984 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 ...
9
votes
1answer
234 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
451 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
votes
1answer
116 views

Intuition for a certain tensor product.

Tensor products occur in lots of places and until recently I thought I understood them at least reasonably well. During the past few weeks, however, I've attended several talks where the tensor ...
8
votes
5answers
980 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
votes
3answers
429 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
votes
2answers
429 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
187 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
votes
3answers
149 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
votes
1answer
907 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
votes
1answer
118 views

Are there any non-obvious colimits of finite abelian groups?

Does the forgetful functor $U : \mathsf{FinAb} \to \mathsf{Ab}$ from finite abelian groups to abelian groups preserve colimits? Morally this should be true, but it is not so easy (for me) to come up ...
8
votes
1answer
159 views

A group whose automorphism group is cyclic

Is there an Abelian group $A$ which is not locally cyclic whose automorphism group is cyclic ?
8
votes
1answer
101 views

Maps to all finite cyclic groups factor implies map to integers factors

Let $G$, $H$ be groups (we lose nothing here if we assume they're abelian), let $f:G\to H$ and $g:G\to \mathbb{Z}$ be homomorphisms. This last map gives us homomorphisms $g_n:G\to {\mathbb{Z}}/{n ...
8
votes
1answer
111 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
votes
1answer
109 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 ...