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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find torsion coefficients of groups

I have to find torsion coefficients of groups $G_1\simeq Z/2\oplus Z/4\oplus Z/3\oplus Z/3\oplus Z/9$ and $G_2\simeq Z/15\oplus Z/20\oplus Z/18$. I want to ask if my calculations are correct. For $...
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1answer
37 views

If $f \otimes \text{id}_{\Bbb Q}$ and $f \otimes \text{id}_{\Bbb{F}_p}$ are isomorphisms, is $f$ an isomorphism?

I would like to know the following "local-global" principle holds (all the tensors are taken over $\Bbb Z$): Let $A,B$ be two abelian groups. Assume that $f \otimes \text{id}_{\Bbb Q}$ and $f \...
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2answers
37 views

Questions about Sylow $p$-groups

Question 1 Is it true that there is only one Sylow $p$-group in an abelian group? Question 2 If there is only one Sylow $p$-group, then it is normal, true? Because if $H\leq G$ is a Sylow $p$-group ...
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2answers
803 views

Abelian Group Element Orders [duplicate]

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 ...
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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 ...
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5answers
510 views

A criterion for a group to be abelian [duplicate]

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 $m$ ...
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1answer
32 views

A sufficient condition for profinite groups

I know that Edwin Hewitt and Kenneth A. Ross (1970) show: Any compact Hausdorff torsion group is profinite. But I don't have the book, the proof seems long and I need only the case of abelian groups ...
3
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1answer
35 views

Existence of open subgroup extending a smaller one

Let $G$ be an abelian topological group and $H \subseteq G$ a dense subgroup (equipped with the subset topology). Furthermore let $V \subseteq H$ be a subgroup that is open in $H$. Does there exist a ...
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0answers
8 views

compact Lie group with non-compact Lie subgroup? [duplicate]

Can there be compact Lie groups with non-compact subgroups? I thought that was not possible until I thought of the torus with the irrational rotations. So if one identifies $U(1)\times U(1)$ with the ...
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1answer
237 views

Arnold's proof of Abel's theorem

I'm seeking help understanding this video. The author considers the equation $ax^5+bx^4+cx^3+dx^2+ex+f = 0$ and shows both the coefficients $a, b$... and solutions $x_1, x_2$... in the complex ...
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1answer
83 views

If $G$ is a locally cyclic group , then is $\operatorname{Aut}(G)$ abelian?

Let $G$ be a locally cyclic group, then is it true that $\operatorname{Aut}(G)$ is abelian ? I know that $G$ has to be abelian but I cannot decide for $\operatorname{Aut}(G)$.
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0answers
8 views

Perfect pairing co-weight lattice and root lattice

Let $\Phi$ be a root system and let $\Lambda_R$ and $\Lambda_W$ denote root lattice and weight lattice. I know that there is a perfect pairing $\Lambda_W \times \Lambda_R^\vee \to \mathbb{Z}$, where $\...
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1answer
28 views

$H \subseteq K$ be subgroups of an infinite abelian group $G$ such that $G/H \cong G/K$ , then are $H,K$ equal or atleast isomorphic?

Let $H \subseteq K$ be subgroups of an infinite abelian group $G$ such that $G/H \cong G/K$ , then is it true that $H=K$ ? Or atleast $H \cong K$ ? ( If $G$ were finite then it would be trivially true ...
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0answers
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Write $\mathbb{Z}^3/L$ as a direct sum of cyclic groups

Let $L\subset \mathbb{Z}$ be the subgroup of $\mathbb{Z}^3$ generated by the elements $(-1,-1,4),(2,4,0),(3,3,8)$. Write $\mathbb{Z}^3/L$ as a direct sum of cyclic groups. I've tried creating a ...
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2answers
29 views

Listing elements of subgroup generated by $\{12,42\}$ in the integers with addition

The subgroup generated by these elements should contain both $12\mathbb{Z}$ and $42\mathbb{Z}$ but also ideals of the form $$ (12k+42j)\mathbb{Z},\;j,k\in\mathbb{Z} $$ Is this the best answer I can ...
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0answers
40 views

Centralizers of Elements in the Free Group

Let $F_n$ be the nonabelian free group on $n$ generators. According to what I have been reading from various sources online, the centralizer of some element $h \in F_n$, denoted as $C_{F_n}(h)$, is an ...
2
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0answers
28 views

Isomorphism of invariant factor decomposition

By the structure theorem, for every finite abelian group $A$, we have an isomorphism $A \cong \mathbb{Z}_{d_1} \oplus \dots \oplus \mathbb{Z}_{d_n}$ for unique $d_i$, s.t. $d_i | d_{i+1}$. My question ...
0
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1answer
57 views

Determine abelian groups with 48 elements

I am just doing some revision for my linear algebra exam, and I came across this problem: Determine all abelian groups (up to isomorphism) with exactly 48 elements. I am not sure I have ever ...
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2answers
36 views

What is $pA$ when $p$ is a prime number and $A$ an abelian group?

Let $A$ be a finite abelian $p$-group. I want to prove that $pA$ is also an abelian finite $p$-group, of order strictly less than the order of $A$. The problem is that I don't even know what does the ...
2
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1answer
29 views

Problem with the proof of abelian finite groups decomposition

I can't understand the proof which says that every finite abelian $p$-group can be written as a direct sum of cyclic $p$-groups. I'm using Lang's book of Algebra. My problem is about the following ...
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2answers
90 views

Homomorphism between a group of exponent $m$ and $\mathbb{Z}/m\mathbb{Z}$

Let $G$ be an abelian group of exponent $m$, where $m\in\mathbb{N}$. Is there always a nontrivial group homomorphism between $G$ and $\mathbb{Z}/m\mathbb{Z}$ ? For example, if we have $G=\mathbb{Z}/m\...
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1answer
185 views

Show from the axioms: Addition in a quasifield is abelian

According to wikipedia a quasifield is an algebraic structure $(Q,+,\cdot)$ such that $(Q,+)$ is a group. (As usual, we denote its identity element by $0$.) $(Q\setminus\{0\},\cdot)$ is a loop. (Its ...
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3answers
113 views

Strange question about free abelian group

I was given the following question for homework, but it makes no sense to me. Let $F$ be a free abelian group over a set $S$ with respect to the function $\varphi \colon S \to F$. Identify the set ...
4
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1answer
100 views

Question about accessibility of category of free abelian groups.

I've read, that the accessibility of the category of all free abelian groups is independent on basic set theory (say ZFC). What is the reason for that? And how can I interpret this result? Does it ...
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1answer
42 views

$G$ be an infinite abelian group such that every proper non-trivial subgroup of $G$ is infinite and cyclic ; then is $G$ cyclic?

Let $G$ be an infinite abelian group such that every proper non-trivial subgroup of $G$ is infinite cyclic ; then is $G$ cyclic ? ( The only characterization I know for infinite abelian groups to be ...
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0answers
49 views

Prove that $G = \{\cos x + t \sin x: x\in\mathbb{R}\}$ is an abelian group under multiplication

where $x$ is the angle Closure is easy - Since $x$ is a real number, its $\cos$ component and $\sin$ component will be a real number. Associative property - I guess it means that $(\cos x + t\sin ...
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1answer
49 views

kernel of product of group homomorphisms

Let $f,g:A \to B$ be group homomorphisms, with $B$ abelian. Then $f\cdot g$ is also a group homomorphism. What can I say about $\ker(f \cdot g)$ in terms of $\ker(f)$ and $\ker(g)$?
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2answers
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Are the groups $\mathbb C^* \times \mathbb R^*$ and $\mathbb R^* \times \mathbb R^*$ isomorphic ?

Consider the groups $\mathbb R^* , \mathbb C^*$ under multiplication , I know that they are not isomorphic ( as one of them is divisible but the other is not ) , my question is : Are the groups $\...
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1answer
59 views

Simple characterization of integers among abelian groups

This is part of an early exercise in Freyd's abelian categories. Let $\mathscr{G}$ be the category of abelian groups. The group of integers is distinguished, up to isomorphism, by the facts that: ...
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2answers
55 views

Are the groups $\mathbb R/ \mathbb Z $ and $ \mathbb R^2 / (\mathbb Z \times \{0\} )$ isomorphic?

Is it true that as groups , $\mathbb R/ \mathbb Z \cong \mathbb R^2 / (\mathbb Z \times \{0\} )$ ? I only know that $\mathbb R \cong \mathbb R^2$ (as groups ) but I can see no way to decide whether ...
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1answer
82 views

Least symmetric group having a certain Abelian group as subgroup

Given an Abelian group $G\simeq\bigoplus_{k}\mathbb Z_{p^{n_k}_{k}}$, where $p_1\leq p_2\leq ...$ are primes, how to calculate the least symmetric group $S_n$ having a subgroup isomorphic to $G$?
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Abelianization of $\mathbb{Z}_2*\mathbb{Z}_3$

Intuitively it has to be $$\text{Ab}(\mathbb{Z}_2*\mathbb{Z}_3)=\mathbb{Z}_2\times\mathbb{Z}_3$$ here is my approach on how to prove it $$\mathbb{Z}_2=P(a\mid a^2),\mathbb{Z}_3=P(b\mid b^3)\Rightarrow ...
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On describing a sort of “well-behaved” subgroups of a free abelian group.

I found this question when I tried to figure out what kind of subgroups of a free abelian group behave just as well as in the finitely generated case. Let $M$ be a free abelian group and $N$ a ...
2
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2answers
1k views

Invertability of Singular 2x2 Matrix with all same real values.

Question: Let set G = { matrix [{a a},{a a}] such that a is real but not 0 } represent the set of 2x2 matrices with same elements of the reals excluding a = 0, show that G is a group under matrix ...
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2answers
33 views

Group endomorphisms of simple abelian groups which do not commute by composition. [closed]

What is an example of group homomorphisms $f,g: M \to M$ where $M$ is a simple abelian group such that $f\circ g \ne g\circ f$ ?
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45 views

Fixed Point of Automorphism group of a Cyclic group Z2XZ2^2 I need the command on GAP

Dear Mathematics Stack Exchange, I have a problem that how to write a command in GAP the automorphism group of finite abelian group and their fixed points. Let Z_pXZ_p2 be cyclic group where p is ...
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2answers
35 views

Which one of the below options is correct?

I think the option $(Q)$ is true since $O(Q/\{-1,1\})= 8/2 = 4 = 2^2$. Since order is $p^2$ thus $(Q)$ option is true. Can anyone suggest about option $(P)$? Thanks
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3answers
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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 ...
3
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1answer
36 views

For which odd positive integer $n$ , is it true that $-1$ is not a positive power of $2$ modulo $n$?

For which odd positive integer $n$ , is it true that $-1$ is not a positive power of $2$ modulo $n$ i.e. $[-1] \ne [2^k] , \forall k >0$ in $\mathbb Z_n$ ? Is there any ( at least sufficient ) ...
3
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2answers
47 views

Subgroups of finite abelian groups.

For every subgroup $H$ of a finite abelian group $G,$ there exists a subgroup $N$ of $G$ such that $G/N \cong H.$ I need to prove this or give a counter example. I am aware of isomorphism theorems ...
3
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0answers
34 views

Decomposition of quotient group of lattices

By the Chinese remainder theorem, we know that $\mathbb{Z}_m \cong \prod_{i=1}^l \mathbb{Z}_{p_i^{k_i}}$, where $m=p_1^{k_1} ... p_l^{k_l}$. Now, let $\Lambda = A(\mathbb{Z}^n) \subseteq \mathbb{Z}^...
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0answers
44 views

Additivity of trace

Let $A$ be a finitely generated abelian group and $\alpha:A\to A$ be an endomorphism. Since $A=A_{free}\oplus A_{torsion}$, we can induce $\bar \alpha:A_{free}\to A_{free}$, i.e. $\bar\alpha$ is a map ...
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1answer
34 views

Order of this group?

It s a stupid question probably but i dont know. It was a little question in a test. The order ( cardinality) of $G= \mathbb{Z}_2 \times \mathbb{Z}_6$. I think it s $12$, the direct product is the ...
4
votes
1answer
62 views

Which group is isomorphic to?

If I have an abelian group $G$ of order $p^n$, how can I decide if it's isomorphic to $\Bbb{Z}_p \times \Bbb{Z}_p \times\ldots \times \Bbb{Z}_p$ ($n$ times) or to $\Bbb{Z}_{p^2} \times \Bbb{Z}_p \...
0
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1answer
18 views

Proof that a transitive permutation group (G, X) with G abelian, is sharply regular

As the title states, the question is the following: Let (G, X) be a transitive permutation group, where G is abelian. Show that (G, X) is "sharply regular". First of all I want to notice that in my ...
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0answers
81 views

Properties of finite abelian group

Let $G$ be a finite abelian group of order $n$ . Then choose the correct statement. a) If d divides n, then there exist a subgroup of $G$ of order $d$ b) If d divides n, then there exist an ...
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1answer
38 views

Abelian Group Structures

How can I determine all the subgroups of a commutative group, write the Hasse diagram, using Frobenius-Stickelberger Theorem and the isomorphism to $\mathbb{Z}_m$ of a cyclic group? In particular, for ...
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1answer
51 views

Example of an abelian group $G$ with $A \le G$ but no $B \le G$ with $G = A \oplus B$.

I just read that if $G$ is an abelian group with subgroup $A$, then we could not always find a subgroup $B$ such that $G = A \oplus B$. I tried to come up with an example, let $G = \mathbb Z^{\mathbb ...
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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?
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1answer
70 views

Is $m\mathbb{Z}\otimes_{\mathbb{Z}}\mathbb{Z}/m\mathbb{Z}\cong 0$?

Since each 'generator' of $m\mathbb{Z}\otimes_{\mathbb{Z}}\mathbb{Z}/m\mathbb{Z}$ has the form $km\otimes_{\mathbb{Z}}\bar{a}=k\otimes_{\mathbb{Z}}m\bar{a}=k\otimes_{\mathbb{Z}}0=0$.