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Questions tagged [abelian-groups]

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

105
votes
4answers
21k 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 $A^n := A\times A\times\ldots\times A$...
50
votes
5answers
9k 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 ...
47
votes
2answers
37k 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 cyclic, ...
35
votes
2answers
4k 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 ...
34
votes
3answers
2k 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?
30
votes
1answer
4k 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 ...
29
votes
13answers
36k views

Prove that if $g^2=e$ for all $g$ in $G$ then $G$ is Abelian.

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 ...
29
votes
1answer
1k 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 difficult ...
27
votes
2answers
6k 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} ,...
25
votes
1answer
4k 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 $\{(e_{\...
25
votes
2answers
781 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 ...
23
votes
6answers
26k 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!
23
votes
3answers
4k 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 ...
23
votes
2answers
7k views

Showing that a finite abelian group has a subgroup of order m for each divisor m of n

I have made an attempt to prove that a finite abelian group of order $n$ has a subgroup of order $m$ for every divisor $m$ of $n$. Specifically, I am asked to use a quotient group-induction ...
22
votes
1answer
13k views

Computing the Smith Normal Form

Let $A_R$ be the finitely generated abelian group, determined by the relation-matrix $$R := \begin{bmatrix} -6 & 111 & -36 & 6\\ 5 & -672 & 210 & 74\\ 0 & -255 &...
22
votes
3answers
13k 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 $...
21
votes
2answers
7k views

Is every quotient of a finite abelian group $G$ isomorphic to some subgroup of $G$?

I'm having difficulty with exercise 1.43 of Lang's Algebra. The question states Let $H$ be a subgroup of a finite abelian group $G$. Show that $G$ has a subgroup that is isomorphic to $G/H$. ...
21
votes
3answers
2k 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 ...
19
votes
2answers
2k 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 like ...
19
votes
2answers
758 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 ...
19
votes
2answers
306 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 there ...
19
votes
2answers
6k views

Proving that a subgroup of a finitely generated abelian group is finitely generated

A question says: Using the isomorphism theorems or otherwise, prove that a subgroup of a finitely generated abelian group is finitely generated. I would say that for a finitely generated abelian ...
18
votes
6answers
2k views

How can I visualise groups in Group Theory

I'm having a hard time grasping groups in Group Theory. Is it ok to visualise them as being sets with the group axioms and a binary operation, intuitively as a venn diagram? and G without * is really ...
18
votes
2answers
730 views

How to recognize a finitely generated abelian group as a product of cyclic groups.

Let $G$ be the quotient group $G=\mathbb{Z}^5/N$, where $N$ is generated by $(6,0,-3,0,3)$ and $(0,0,8,4,2)$. Recognize $G$ as a product of cyclic groups. Honestly, I do not know how to solve these ...
17
votes
2answers
3k 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 reads:...
17
votes
2answers
4k 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 ...
17
votes
3answers
1k views

A group such that $a^m b^m = b^m a^m$ and $a^n b^n = b^n a^n$ ($m$, $n$ coprime) is abelian?

Let $(G,.)$ be a group and $m,n \in\mathbb Z$ such that $\gcd(m,n)=1$. Assume that $$ \forall a,b \in G, \,a^mb^m=b^ma^m,$$ $$\forall a,b \in G, \, a^nb^n=b^na^n.$$ Then how prove $G$ is an abelian ...
16
votes
3answers
680 views

infinite abelian group where all elements have order 1, 2, or 4

Let $A$ be a (not necessarily finitely generated) abelian group where all elements have order 1, 2, or 4. Does it follow that $A$ can be written as a direct sum $(\bigoplus _\alpha \mathbb Z/4) \oplus ...
16
votes
3answers
1k 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. ...
15
votes
4answers
7k 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?
15
votes
5answers
407 views

A curiosity: how do we prove $\mathbb{R}$ is closed under addition and multiplication?

So I tried looking around for this question, but I didn't find much of anything - mostly unrelated-but-similarly-worded stuff. So either I suck at Googling or whatever but I'll get to the point. So ...
15
votes
4answers
14k views

Proof that all abelian simple groups are cyclic groups of prime order

Just wanted some feedback to ensure I did not make any mistakes with this proof. Thanks! Since $G$ is abelian, every subgroup is normal. Since $G$ is simple, the only subgroups of $G$ are $1$ and $G$,...
15
votes
2answers
3k 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 ("...
15
votes
1answer
673 views

Is there a good example of a subgroup of an infinitely generated abelian group that is not isomorphic to a quotient of that group?

Whilst I understand the classification of the finitely generated abelian groups, this had me wondering whether there is a subgroup $H$ of a general (necessarily infinitely generated) abelian group $G$ ...
15
votes
2answers
2k views

Rank-nullity theorem for free $\mathbb Z$-modules

From linear algebra we know that given vector spaces $V$, $W$ over a field $k$ and a linear map $f\colon V\to W$ we have $$\dim V = \dim \operatorname{im} f + \dim \ker f.$$ Is this still true when ...
15
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 ...
14
votes
4answers
2k views

Show that a nonabelian group must have at least five distinct elements [closed]

Show that a nonabelian group must have at least five distinct elements. I just learn abstract algebra by self study. I want help to solve this problem. Just give me a hint.
14
votes
5answers
1k views

Prove that $(a_1a_2\cdots a_n)^{2} = e$ in a finite Abelian group

Let $G$ be a finite abelian group, $G = \{e, a_{1}, a_{2}, ..., a_{n} \}$. Prove that $(a_{1}a_{2}\cdot \cdot \cdot a_{n})^{2} = e$. I've been stuck on this problem for quite some time. Could someone ...
14
votes
1answer
789 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) ...
13
votes
1answer
344 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 ...
13
votes
2answers
739 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 ...
13
votes
2answers
150 views

Can one distinguish finite groups by their maps from abelian groups?

Given a finite group $G$, is $G$ determined by its category of maps from abelian groups? Specifically, we can form the category $G_A$ of "abelian points" of $G$ with objects pairs $(A,\phi)$, with $\...
12
votes
4answers
2k views

Non-abelian group with infinitely many abelian subgroups

I'm looking for a non-abelian group which has infinitely many abelian subgroups. Do you know any examples of such groups?
12
votes
7answers
14k views

Give an example of a noncyclic Abelian group all of whose proper subgroups are cyclic.

I've tried but I could not find a noncyclic Abelian group all of whose proper subgroups are cyclic. please help me.
12
votes
2answers
533 views

Does an abelian subgroup inject into the abelianisation of the whole group? [closed]

If $H <G $ are groups and H is abelian, do we get an injection from H into $G/[G,G] $?
12
votes
1answer
4k views

Showing that a cyclic automorphism group makes a finite group abelian

From a bank of previous masters exams: Let $G$ be a finite group such that its automorphism group $\operatorname{Aut}(G)$ is cyclic. Prove that $G$ is abelian. Here's what I was thinking. Let $\...
12
votes
4answers
2k 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 ...
12
votes
2answers
732 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.
12
votes
2answers
2k views

Show that any abelian transitive subgroup of $S_n$ has order $n$

Can anybody tell me what is known about the classification of abelian transitive groups of the symmetric groups? Let $G$ be a an abelian transitive subgroup of the symmetric group $S_n$. Show that $...
12
votes
1answer
160 views

Is there a surjective morphism from an infinite direct product of copies of $\mathbb{Z}$ to an infinite direct sum of copies of $\mathbb{Z}$?

Is there a surjective morphism $\mathbb{Z}^I\to \mathbb{Z}^{(J)}$ for some $I,J$? i) I'm asking about group morphisms ii) $\mathbb{Z}^{(J)}$ denotes the direct sum of $J$ copies of $\mathbb Z$ iii) ...