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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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
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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 ...
12
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4answers
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 ...
12
votes
3answers
430 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. ...
18
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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
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2answers
913 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 ...
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4answers
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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 ...
19
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1answer
851 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 ...
12
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3answers
290 views

How to prove that a group with some properties is abelian?

Let $(G,.)$ be a group and $m,n\in\mathbb Z$ such that $\gcd(m,n)=1 $ and $$ \forall a,b \in G:a^mb^m=b^ma^m$$ $$\forall a,b \in G:a^nb^n=b^na^n.$$ Then how prove $G$ is an abelian group ? Thanks ...
12
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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} ...
50
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
706 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
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1answer
219 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 ...
28
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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 ...
7
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
935 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 ...
5
votes
1answer
318 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.
2
votes
3answers
524 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 $5$ , ...
6
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2answers
398 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 ...
10
votes
4answers
497 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 ...
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 ...
13
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2answers
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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
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1answer
192 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
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2answers
615 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 ...
8
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 ...
7
votes
1answer
839 views

Equivalences and isomorphisms of short exact sequences

In case it's necessary, I'm working in the category $\mathbf{Ab}$ of abelian groups. My question concerns what I find to be a strange way of viewing the elements of the Ext group $\mbox{Ext}(A,B)$ of ...
6
votes
2answers
498 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 ...
6
votes
6answers
5k 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.
4
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2answers
129 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 $A$ ...
27
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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?
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
851 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
185 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
329 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
584 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
327 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
273 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
274 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
140 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
924 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
758 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
189 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} ...
4
votes
3answers
135 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
520 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
272 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
115 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
605 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
106 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 .
2
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1answer
296 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 ...
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4answers
1k 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.