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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6
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1answer
80 views

About translating subsets of $\Bbb Z.$

This is a continuation of About translating subsets of R2. Is it possible to find a pair of sets $A,B\subseteq\Bbb Z$ such that A is a union of translated (only translations are allowed) copies of ...
1
vote
1answer
40 views

Are these exactly the abelian groups (2)?

This is a continuation of Are these exactly the abelian groups? I would like to consider another condition on a group and see if it implies commutativity. The condition is $$(\forall A,B\subseteq ...
0
votes
1answer
41 views

Under what conditions can the symmetric group be isomorphic to the abelian group?

The symmetric group is the set of all permutations. My question addresses the representability of the symmetric group using only additions. I am guessing that on the finite field $\mathbb{Z}/n ...
1
vote
2answers
25 views

Rank($U$) = Rank($U^2$) for group of units $U$

I am reading the paper "Algebraic Integers on the Unit Circle" by Ryan C. Daileda (http://www.sciencedirect.com/science/article/pii/S0022314X05002027). I am confused about how he concludes that the ...
2
votes
1answer
385 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
45 views

How to count the number of elements of given order?

I am trying to prove the following result. Let $G$ and $G'$ be two finite abelian groups. Besides, they have the same number of elements of any given order. Prove that $G\cong G'$. My attempt is ...
2
votes
4answers
148 views

there is no injective group homomorphism from $\mathbb Z\times\mathbb Z$ into $\mathbb Z$

there is no injective group homomorphism from $\mathbb Z\times\mathbb Z$ into $\mathbb Z$ But i don't know why it is true. should i investigate all group homomorphisms from $\mathbb Z\times\mathbb ...
7
votes
1answer
149 views

Let G be a group of order $n$, where $n$ is a positive integer relatively prime to $\varphi(n)$. Show that G is cyclic.

Let G be a group of order $n$, where $n$ is a positive integer relatively prime to $\varphi(n)$. Show that G is cyclic. You may only assume the Feit-Thompson theorem here and prove in the following ...
3
votes
3answers
148 views

Basic Group Theory question

This is not so much a plea of ignorance, but rather me trying to see whether intuitively I actually understand what is going on in group theory. The question asks What group is ...
3
votes
1answer
234 views

normal p-subgroups of a finite group and chief factor

Let $G$ be a finite solvable group. Let $K/H$ be a chief factor of $G$ that is not of prime order, where $K$ is a $p$-subgroup of $G$ for some prime $p$ divides the order of $G$. Let $S$ be a proper ...
0
votes
0answers
45 views

Finite generated abelian group $G$ and $H<G$. What is the rank of $(G/H)/(G/H)_t$?

I saw another question about this problem here. However there are quite different answers from my expectation. Anyway, here are my trials. Trial 1 : By structure theorem, $G\cong G_t\oplus F_1$ ...
0
votes
1answer
45 views

generators of groups from exact sequence

Suppose I have a middle term exact sequence of finitely generated abelian groups $G \longrightarrow H \longrightarrow K$. How do I get the generators of $H$ if I know the same for other two groups?
0
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3answers
73 views

A question on Abelian Groups

Prove that every subgroup of an Abelian group is Abelian but the converse is not true. I recently stumbled onto this question , but not able to solve it . Please help me out!
2
votes
1answer
47 views

If direct limits of matrices are isomorphic, is the direct limit of the transpose matrices also isomorphic?

On the one hand, the following conjecture seems reasonable, but on the other hand it doesn't seem natural because some objects are being dualised while others are not. I would appreciate if anyone ...
7
votes
1answer
470 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 ...
5
votes
1answer
96 views

Additive non-abelian group?

Sometimes I see in books the term "additive abelian groups". In my opinion, when we use addition to represent the group operation, we already have in mind that the operation is commutative. So ...
3
votes
5answers
87 views

Proof that a group is abelian.

If $(G,*)$ is a group and $(a * b)^2 = a^2 * b^2$ then $(G, *)$ is abelian for all $a,b \in G$. I know that I have to show $G$ is commutative, ie $a * b = b * a$ I have done this by first using ...
1
vote
1answer
46 views

Commutators Calculus

I was trying to understand the above Corollary but I have a problem, namely why in the second to last line $A_0 \leq \zeta_p(G)$? Any ideas? Definitions By recurrence we define $[x,_0\, y]=x$; ...
0
votes
1answer
98 views

Let $G$ be a finite abelian group and let $p$ be a prime that divides order of $G$. then $G$ has an element of order $p$

Let $G$ be a finite abelian group and let $p$ be a prime that divides order of $G$. then $G$ has an element of order $p$ Proof When $G$ is abelian. First note that if $|G|$ is prime, then $G \approx ...
1
vote
4answers
65 views

Proving that a group $(G, \ast)$ is abelian if $x^3=x$ for all $x\in G$

If $(G, \ast)$ is a group so that $x^3=x$ for all $x\in G$ then $G$ is abelian
4
votes
2answers
575 views

Order of products of elements in a finite Abelian group

We want to show that if $a,b\in G$ where $G$ is a finite Abelian group, we have $\operatorname{LCM}(|a|,|b|) = |ab|$ given that $ab \neq e$. How I approached this question was by saying let ...
4
votes
0answers
41 views

Commutativity of direct and inverse limits

In exercise 5.34(iv) of Homological Algebra book by Rotman one is asked to prove that direct limits and inverse limits do not necessarily commute. I have two questions : 1.) Is it true that ...
1
vote
1answer
38 views

A group with bounded element orders and its minimal and maximal subgroups.

Let $n>1$ be an integer. Is there an abelian group $G$ with all elements of order less than $n$ for which exactly one of these conditions is correct: 1) every non-trivial subgroup of $G$ contains ...
4
votes
2answers
108 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 ...
0
votes
0answers
15 views

maximal and minimal subgroups of torsion abelian groups

Is there a torsion abelian group $G$ for which exactly one of these conditions is correct: 1) every non-trivial subgroup of $G$ contains a minimal (non-trivial) subgroup of $G$. 2) every proper ...
7
votes
3answers
401 views

Abelian Group Element Orders

I want to show that if a finite abelian group has elements of order $m$ and $n$ then it will have an element of order $\text{lcm}(m,n)$. First I proved the lemma if $a$ has order $m$ and $b$ has ...
0
votes
2answers
50 views

What can I say about the quotient group?

Let $G$ be a group of order $24$, and let $H$ be a normal subgroup of order $6$. So the quotient group $ {G\over H} $ is Abelian group?. What can I say about the quotient group beside her order?
10
votes
2answers
807 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 ...
22
votes
2answers
934 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?
1
vote
0answers
23 views

Automorphism group of an abelian p-group

I'd like to know if it's known the structure of the automorphism group of an abelian $p$-group with the minimal condition on subgroups, for some prime number p. I know that if $A$ is an abelian ...
4
votes
4answers
65 views

$n$th power map is an automorphism implies abelian group?

If $G$ is a finite group and $\phi(x) = x^n$ is an automorphism of $G$ does this imply $G$ is abelian? I've been reading this page. Def: A group $G$ is said to be $n$-abelian if $(ab)^n=a^nb^n$ ...
0
votes
3answers
77 views

Finding an Abelian subgroup

I am working on my homework and am currently stuck at the following question: Let $\mathcal S_{10}$ be the group of all permutations of the set of elements {1, 2, ..., 10}. Find an Abelian ...
7
votes
1answer
114 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 / ...
1
vote
2answers
37 views

A finite $\mathbb{Z}$-module whose submodules are totally ordered by inclusion.

I have the following problem: Let $M$ be a finite $\mathbb{Z}$-module such that set of the submodules is totally ordered by inclusion. Prove that there exist a prime $p$ such that $|M|=p^\alpha$ ...
2
votes
1answer
37 views

Show that, given an element $\bar{x}\in\mathbb{Q}/\mathbb{Z}$, there is an integer $n \ge 1$ such that $n\bar{x} = 0$. [duplicate]

Consider $\mathbb{Z}$ as a subgroup of the additive group $\mathbb{Q}$ of rational numbers. >Show that, given an element $\bar{x}\in\mathbb{Q}/\mathbb{Z}$, there is an integer $n \ge 1$ such that ...
2
votes
1answer
48 views

How to prove that $\mathbb Z/p$ is not a direct summand of any direct sum of copies of $\mathbb Z/n$?

How can I prove that $\mathbb Z/p$ ($p$ is a prime) cannot be a direct summand of any arbitrary direct sum of copies of $\mathbb Z/n$, where $p^2$ divides $n$?
3
votes
2answers
222 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} ...
2
votes
1answer
43 views

$\operatorname{Hom}(G,\Bbb C\setminus \{0\})$ non- isomorphic to $\operatorname{Hom}(G, \Bbb T)$?

Do you have an example of an abelian group $G$ for which $\operatorname{Hom}(G,\Bbb C\setminus \{0\})$ is not isomorphic to $\operatorname{Hom}(G, \Bbb T)$? $\Bbb C$ is the complex plane and $\Bbb ...
2
votes
3answers
653 views

Prove that a finite abelian group is simple if and only if its order is prime.

So I'm having trouble with this problem. I know that the definition of a simple group means that the group has no nontrivial subgroups. I know that this can be proven somehow with the help of the ...
5
votes
1answer
41 views

Let $G$ be a group and $a,b,c \in G$. Given that $abc$ and $cba$ are conjugated, prove that $G$ is abelian.

Let $G$ be a group and $a,b,c \in G$. Given that $abc$ and $cba$ are conjugated, prove that $G$ is abelian. In other words, if for any $a,b,c \in G$ there is a $g \in G$ so that $a b c = g c b a ...
2
votes
2answers
60 views

The factor group $\mathbb{R}^{*}/\{-1,1\}$ is isomorphic to $\mathbb{Z}_{2}$. True or False.

I have been told that the answer for this question is False. And I'm trying to understand why. For what I understand so far, $\mathbb{Z} _2$ is an abelian group. I also understand that for a group ...
2
votes
1answer
74 views

What we know about $\mathbb{Q}/\mathbb{Z}$ as a group? [closed]

What we know about $\mathbb{Q}/\mathbb{Z}$ as a group? Are there any interesting properties? Best regards.
0
votes
1answer
45 views

Self-dualiy of the subgroup lattice of finite abelain groups

For each abelain finite group $G$ let $\mathcal L(G)$ be the lattice of all subgroups of $G$. For which abelian finite groups $G$, is $(\mathcal L(G),\subseteq)$ order-isomorphic to $(\mathcal ...
0
votes
0answers
58 views

Free groups vs. free abelian groups

I'm trying to solve this question in page 74 of Hungerford's book: A free abelian group is a free group (Section I.9) if and only if it is cyclic. I have no idea how to proceed, a solution or a ...
1
vote
1answer
19 views

Proof about existence of subgroups in finite abelian group

I am trying to understand the proof of the following theorem. Theorem: If $G$ is a finite abelian group and $d$ is a divisor of $|G|$, then $G$ contains a subgroup of order $d$. Proof: Let ...
3
votes
2answers
50 views

$mA = 0 = nC, \ \gcd(m,n) = 1 \Rightarrow $ every extension of $A$ by $C$ splits

This is Exercise 7.14(ii) from Rotman, Introduction to homological algebra, and I'm stuck on it. If $A$ and $C$ are abelian groups, with $mA = 0 = nC $ and $\gcd(m,n) = 1$ then every extension of ...
1
vote
2answers
528 views

What does it mean for a group to be Abelian?

I'm revising questions on groups for exams, and I still can't quite understand what an Abelian group is. Please help me understand, if anyone could give me a more simple explanation.
3
votes
1answer
60 views

Normal Abelian Subgroup does not imply Abelian Quotient Group

I'm a bit confused and just need some clarification about what I am missing in this: I have $S_4$ with normal subgroup $N=\lbrace(),(1,2)(3,4),(1,3)(2,4),(1,4)(2,3)\rbrace$. I know that $N$ is ...
0
votes
0answers
41 views

the torsion subgroup of E(Q) (eliptic curves)

if $E$ is an elliptic curve over $Q$, then why $E(Q)_{\rm tor}$ is group and finite set ?
4
votes
2answers
119 views

Tensor product of a finitely generated abelian group and the field of rational numbers

Let $G$ be a a finitely generated abelian group. Then $G\otimes_\mathbb{Z} \mathbb{Q} = 0$ if and only if $G$ is a finite group. The "if" part is easy. The "only if" part can be proved using the ...