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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3
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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 \...
0
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2answers
38 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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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 ...
0
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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 ...
0
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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 $\...
1
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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
27 views

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 ...
4
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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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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 ...
2
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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 ...
0
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2answers
37 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
votes
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 ...
7
votes
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 ...
3
votes
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\...
1
vote
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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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 ...
2
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2answers
88 views

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
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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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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0answers
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 ...
3
votes
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
votes
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 ...
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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 ...
1
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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 \...
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}^...
0
votes
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 ...
1
vote
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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4answers
1k 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?
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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 $km\otimes_{\mathbb{Z}}\bar{a}=k\otimes_{\mathbb{Z}}m\bar{a}=k\otimes_{\mathbb{Z}}0=0$.
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0answers
29 views

Elementary Abelian p groups

How to show that if any group of class 2 has only two conjugacy class sizes, then its centre and quotient by commutator both are elementary abelian?
2
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1answer
45 views

Order of subgroup generated by two cyclic subgroups in $S_6$.

Let $S_6$ be the symmetric group, and $\alpha=(13456)$ and $\beta=(132)$ be its two permutations. How can we find the order of the subgroup generated by $\alpha$ and $\beta$. SOl: $\alpha^5$=...
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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1answer
33 views

Quotient group of free abelian group

Let $A$ be a free abelian group, i.e. $A=\bigoplus_\alpha \mathbb Z$. Also let $B$ be a subgroup of $A$. Prove that $A/B\cong\mathbb Z$ implies $A=B\oplus \mathbb Z$. p.s. Actually this appears in ...
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0answers
39 views

adding a relation to a group

We start with a group $G$ such that there is an abelian subgroup $\mathbb{Z}^3(e,f,g) \subset G$. Assume we have $G/\left\{ g = 1 \right\} \cong \mathbb{Z}^2(e,f)$. What can one say about $G$, except ...
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vote
1answer
34 views

Proving that the set of positive integers does not form a group under addition

The set of positive integers under addition has closure because it goes on for infinity and you will always have the element $a + b$. I am also aware that addition is associative but do we include $0$...
1
vote
1answer
65 views

Are all complete groups abelian?

Hi: Let $G$ be a group and $G'$ it's commutator subgroup. Then $G > G' > 1$ is a series of normal subgroups of $G$. Suppose $G$ is complete. Then, if I'm not wrong, $Aut(G)$ is the stabilizer of ...
0
votes
1answer
28 views

multiple of a crossed homomorphism from finite group to a divisible one is principal

Let $\pi$ be a finite group, $\left|\pi\right|=n$ , acting on an abelian, torsion-free, $n$ -divisible group $D$ (i.e., every element of $D$ is divisible by $n$ ). Consider a crossed ...
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0answers
60 views

Can you prove that modules over a field are vector spaces in a categorical way?

Let $\mathcal{C}$ be the category of abelian groups endowed with tensor product. It is a monoidal category with unit object the integers. Given a monoid $(R, \mu_R , \iota_R)= R$ in this category (i.e....
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0answers
16 views

Isomorphism classes and invariant factors of abelian group

Let $G$ be an abelian group with $ord(G)=3374=2\cdot 7\cdot 241$. Calculate all isomorphism classes with the invariant factors $k_1\ ...k_n$ sucht that $k_i$ divides $k_j$ $(i<j)$. Since $ord(G)=...
0
votes
1answer
34 views

When this map $f(x)=x^2$ will be a automorphism on $G$, where $G$ is a commutative group.

I know that for a commutative group $G$ the map $f(x)=x^2$ is a homomorphism from $G$ to $G$. My question : When this map $f$ will be a automorphism on $G$, where $G$ is a commutative group. In other ...
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2answers
100 views

If $a^3=1$, is $G$ abelian?

If $G$ is a group that satisfies $a^3=1$ for every $a\in G$, then is $G$ abelian? This is an exercise I found in Jacobson's Basic Algebra. It is analogous to the question: If $G$ is a group that ...
3
votes
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 ...