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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11
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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
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 ...
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 ...
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. ...
17
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1answer
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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 ...
7
votes
2answers
865 views

Product of elements of a finite abelian group

Suppose $G=\{a_1,...,a_n\}$ is a finite abelian group, and let $x=a_1a_2\dotsm a_n$. Prove that if there is more than one element of order $2$ then $x=e$. What I've done so far: (#1 is just for ...
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} ...
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 ...
22
votes
1answer
696 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 ...
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 ...
19
votes
1answer
818 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 ...
1
vote
4answers
2k views

Need to prove that (S,*) defined by the binary operation a*b = a+b+ab is an abelian group on S = R \ {1}

So basically this proof centers around proving that (S,*) is a group, as it's quite easy to see that it's abelian as both addition and multiplication are commutative. My issue is finding an identity ...
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 ...
6
votes
1answer
5k views

Computing the Smith Normal Form

This question is related to the Smith Normal Form of Matrices: Let $A_R$ be the finitely generated abelian group, determined by the relation-matrix $R :=$ $$ \begin{bmatrix} -6 & 111 & ...
8
votes
1answer
910 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 ...
2
votes
3answers
495 views

On Groups of Order 315 with a unique sylow 3-subgroup .

in dummit and foote , an exercise asked me to prove that , if$ G$ is a group of order $315$ , $G$ has a normal sylow $3$-subgroup then , $G$ is abelian . this is exercise number $27$ , section $5$ , ...
4
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1answer
346 views

abelian transitive subgroups

Can anybody tell me what is known about the classification of abelian transitive groups of the symmetric groups? For instance: Let $G$ be a an abelian transitive subgroup of the symmetric group ...
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 ...
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 ...
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 ...
13
votes
2answers
576 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 ...
6
votes
1answer
2k 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 ...
4
votes
1answer
290 views

Let $G$ be a group, where $(ab)^3=a^3b^3$ and $(ab)^5=a^5b^5$. Prove that $G $ is an abelian group.

Let G be a group, where $(ab)^3=(a^3)(b^3)$ and $(ab)^5=(a^5)(b^5)$. Prove that $G$ is abelian group. Thank you in advance. Any help is appreciated.
6
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6answers
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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.
4
votes
2answers
125 views

Finite Abelian groups with the same number of elements for all orders are isomorphic [closed]

Let $A$ and $B$ be finite abelian groups. Suppose that for every natural number $m$, the number of elements of order $m$ in $A$ is equal to the number of elements of order $m$ in $B$. Prove that $A$ ...
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?
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 ...
7
votes
3answers
1k views

Finite abelian $p$-group with only one subgroup size $p$ is cyclic

My goal is to prove this: If $G$ is a finite abelian $p$-group with a unique subgroup of size $p$, then $G$ is cyclic. I tried to prove this by induction on $n$, where $|G| = p^n$ but was not ...
5
votes
2answers
846 views

If a group is $3$-abelian and $5$-abelian, then it is abelian

In a group $(Z,*)$, $(a*b)^{5}=a^{5}*b^{5},\forall a,b\in Z$ and $(a*b)^{3}=a^{3}*b^{3}$ then prove that $Z$ is abelian. I know that for three consecutive integer if $(a*b)^{i}=a^{i}*b^{i},\forall ...
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^ ...
6
votes
2answers
326 views

Whence this generalization of linear (in)dependence?

I recently came across a definition of (in)dependence that is supposed to be a generalization of linear (in)dependence among a set of vectors: An element $x$ is dependent on a set of elements ...
5
votes
2answers
567 views

A condition for a subgroup of a finitely generated free abelian group to have finite index

Let A be a free Abelian group of finite rank and B be a subgroup of A such that $A=B+pA$ for some prime number p, then how to prove $B$ is a subgroup of finite index in A? And if $A=B+pA$ holds for ...
3
votes
2answers
304 views

Direct limit of $\mathbb{Z}$-homomorphisms

What is the direct limit of the following sequence of $\mathbb{Z}$-homomorphisms (as groups)? $$ \mathbb{Z} \xrightarrow{2} \mathbb{Z} \xrightarrow{3} \mathbb{Z}\xrightarrow{5} ...
7
votes
3answers
271 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 ...
5
votes
3answers
273 views

$G$ group, $H \trianglelefteq G$, $\vert H \vert$ prime, then $H \leq Z(G)$

Let $G$ be a finite group. Let $H \trianglelefteq G$, with $\vert H \vert = p$, a prime, where $p$ is the smallest prime dividing $\vert G \vert$. Prove that $H \leq Z(G)$. (Hint: If $a \in H$, by ...
3
votes
1answer
139 views

When is a divisible group a power of the multiplicative group of an algebraically closed field?

It is known that for any algebraically closed field $\mathbb{F}$ its multiplicative group $\mathbb{F}^*$ is a divisible group, and consequently any power $\mathbb{F}^*\times\cdots\times \mathbb{F}^*.$ ...
1
vote
1answer
887 views

Subgroups of a finitely generated abelian group without torsion

If $G\cong \mathbb{Z}\times \mathbb{Z}\times \dots \times \mathbb{Z}$ is a finitely generated abelian group without torsion of rank $n$, where $n$ is the number of copies of $\mathbb{Z}$. Then any ...
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2answers
734 views

Non-isomorphic abelian groups of order $19^5$

I am trying to classify abelian groups of order $19^5$ up to isomorphism. Can anyone provide any approaches or hints?
6
votes
2answers
432 views

Simple proof of the structure theorems for finite abelian groups

Many proofs of the structure theorems for finite abelian groups first reduce to the problem to $p$-groups, which is fine and is an important technique. However, it seems to me that a simple proof can ...
4
votes
3answers
124 views

Examples of loops which have two-sided inverses.

Are there any neat examples of non-associative loops such that for each element a in the loop there exists $a^{-1}$ so that $a*a^{-1}=1=a^{-1}*a$. Even cooler would be a commutative loop. Also: are ...
3
votes
2answers
502 views

Finitely generated abelian groups: If $G \times K$ is isomorphic to $H \times K$, then $G$ is isomorphic to $H$.

Let $G,H,K$ be finitely generated abelian groups. If $G \times K$ is isomorphic to $H \times K$, then $G$ is isomorphic to $H$. What I have thought is that fundamental theorem of abelian groups can ...
3
votes
3answers
271 views

Check if $(\mathbb Z_7, \odot)$ is an abelian group, issue in finding inverse element

Take $\mathbb Z_7$ and the operation $\odot$ defined on it as follows $\forall a,b \in \mathbb Z_7$: $$\begin{aligned} a \odot b=a+b+3\end{aligned}$$ Check if $(\mathbb Z_7, \odot)$ is a group and ...
2
votes
1answer
114 views

Find a torsion free, non cyclic, abelian group $A$ such that $\operatorname{Aut}(A)$ has order 2

Is there any chance to find a torsion free, non cyclic, abelian group $A$ such that $\operatorname{Aut}(A)=\mathbb Z_2$? ($\mathbb Z_2$ is the cyclic group of order $2$) Notation ...
2
votes
1answer
587 views

Subgroups and quotient groups of finite abelian $p$-groups

Motivation The fundamental theorem of finite abelian groups gives us a concise description of the isomorphism types of finite abelian $p$-groups $G$ (in the following, $p$ is a fixed prime). The ...
2
votes
2answers
105 views

Show that the mapping $x\rightarrow x^{-1}$ of $G$ onto $G$ is an isomorphism iff $G$ is abelian

Question : Show that the mapping $x\rightarrow x^{-1}$ of $G$ onto $G$ is an isomorphism iff $G$ is abelian, $x\in G $ I need some pointers for proving this .
1
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4answers
916 views

Why is the set of natural numbers considered a non-Abelian group?

I don't understand why the set of natural numbers constitutes a commutative monoid with addition, but is not considered an Abelian group.
1
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1answer
289 views

Computing Quotient Groups $\mathbb{Z}_4 \times \mathbb{Z}_{10} / \langle (2, 4) \rangle$, $\mathbb{Z} \times \mathbb{Z}_{6}/ \langle (1, 2) \rangle$

Let $G/H = \mathbb{Z}_{4} \times \mathbb{Z}_{10} / \langle (2, 4) \rangle$. I know that $|G/H|$ = 4, so $G/H \simeq \mathbb{Z}_{2} \times \mathbb{Z}_{2}$ or $\mathbb{Z}_{4}$. Since $G/H$ has an ...
1
vote
5answers
405 views

A criterion for a group to be abelian

I noted a discussion on groups being abelian under a certain restriction on powers of elements, e.g. http://tiny.cc/chs45. Maybe this result (probably not too well-known) concludes it all. Let and ...
7
votes
1answer
859 views

Isomorphism Types

I'm a little panicking right now. I have finals soon, and I don't know how to go about solving this: Classify the isomorphism types of abelian groups of order 44. Solutions or even hints would be ...
6
votes
2answers
182 views

Computing easy direct limit of groups

How do I start computing easy direct limit of groups: 1) $\mathbb{Z} \overset{1}\longrightarrow \mathbb{Z} \overset{2}\longrightarrow \mathbb{Z} \overset{3}\longrightarrow \mathbb{Z} ...