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Questions tagged [abelian-groups]

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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3answers
177 views

Prove that every group of order 15 is abelian? [duplicate]

I had seen this proof at many places, but everywhere sylows theorem is used. So is their any way to solve it without using sylows theorem?
2
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2answers
36 views

Inductive Proof of Group with Prime Decomposition is Isomorphic to Direct Product of Cyclic Groups

My lecturer set as a bonus exercise the following induction proof: If $G$ is a finite abelian group $|G| = p_1^{n_1} \cdots p_s^{n_s}$ is the decomposition of $|G|$ into a product of distinct prime ...
3
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2answers
93 views

$G$ is an abelian group, and a permutation $T$. Suppose that $x-T(x)\neq y-T(y)$ for all $x\neq y$. Show that $p(x)=x-T(x)$ is also a permutation

Let $G=(A,+)$ be an abelian group, T a permutation on $A$. Assume that $x - T(x) \neq y - T(y)$ for all distinct $x,y \in A$. Show that $p(x)=x - T(x)$ is a permutation on $A$. For that I would show ...
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2answers
82 views

Let $G$ be a non-abelian group of order $10$. Prove that G has a trivial center.

I have done this as follows: Let $G$ be a non-abelian group of order $10$. If possible let a non-identity element say $a \in G$ is in $Z(G)$. Now by lagrange's theorem , $|a|= 2,5$ or $10$. If $|a|...
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0answers
28 views

How to construct a homomorphism from an abelian group to $\mathbb{Q}$ [duplicate]

For any abelian group $A$. Is it possible to construct a nontrivial homomorphism from $A \to \mathbb{Q}$? Is it possible if $A$ is a finite generate abelian group?
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46 views

Subgroups of the integers

Want to show: Consider the additive group, $\mathbb{Z}$. Show that the subgroups are of the form $n\mathbb{Z}$, for some $n\in \mathbb{Z}$ Proof: Note that for $1\in \mathbb{Z}$, $\langle 1\...
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1answer
23 views

Injection of $\mathbb{Z}$ into a p-adically complete abelian group

If $M$ is a p-adically complete abelian group, so that it's a $\mathbb{Z}_p$-module, and we have an injective homomorphism $\phi: \mathbb{Z} \hookrightarrow M$, is it true that the induced ...
0
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1answer
23 views

Cardinality of an intersection of two submodules.

Assume $p$ is a prime number and $q = p^2$. Denote by $A$ the ring $\mathbb{Z} / q \mathbb{Z}$. Consider a finite type module $M$ over $A$ whith cardinality $q^N$ where $N$ is an integer, $N>0$. ...
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0answers
18 views

Determining the Cyclic Decomposition of a Finitely Generated Abelian Group given a set of relations.

I am given a group $G=\text{Span}(w,x,y,z)$ with relations defined by $$\begin{bmatrix}0&0&1&3\\-2&1&1&3\\-2&4&1&3\\0&-3&1&5\end{bmatrix}\begin{bmatrix}...
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2answers
98 views

If every nonidentity element in a group is of order $2$, the group is abelian [duplicate]

Let $G$ be a group. Show that if every non-identity element in $G$ has order $2$ then $G$ is abelian. Proof: Let $a,b $ be non-identity elements in $G$. Since $|a|=|b|=2$ , that means $ab=babaab$ $=$...
2
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1answer
42 views

Groups, subgroups and quotient groups

(Q,+) abelian group, (Z,+) his subgroup and Q/Z a quotient group. Find all subgroups of <$\frac{\hat1}{6}$>,$\frac{\hat1}{6}$ $\in$ Q/Z. The elements of <$\frac{\hat1}{6}$> are { $\frac{\hat1}{...
2
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2answers
63 views

Why not $\mathbb{Q}$ and $\mathbb{Q}\times \mathbb{Q}\cong \mathbb{Q}^2$ not isomorphic as additive groups?

We know that $\mathbb{R}$ and $\mathbb{R}^2$ are isomorphic as additive groups(as they are isomorphic as $\mathbb{Q}$ vector spaces) under the axiom of choice. But why not $(\mathbb{Q}, +)$ and $(\...
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1answer
42 views

additive order of any zero divisor in $Z_{p^2}$ is p, is it true?

This result was used in a proof of a theorem, i am not sure if it's true. can someone tell the proof idea. Can it be generalized to additive order of any zero divisor in $Z_{p^k}$, is there any ...
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1answer
37 views

What is the maximal order among elements in $\mathbb{Z}\times(\mathbb{Z}/10\Bbb{Z})/\langle(5,4)\rangle$?

Here's a problem about a group : $\mathbb{Z}\times(\mathbb{Z}/10\Bbb{Z})$ Problem : Find the maximal order of elements of $\mathbb{Z}\times(\mathbb{Z}/10\Bbb{Z})/\langle(5,4)\rangle$ The answer is......
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1answer
21 views

A divisibility property of the direct sum of abelian groups

I'm reading a book and the author leaves as an exercize the proof of the following, assuming that $n\ge 1$ is an integer and $b\in B,c\in C$ If $A=B\oplus C$, then $n\mid (b+c)$ iff $n\mid b$, $n\...
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0answers
27 views

Let $f(z)=|z-1|/(|z|^p+1)^{1/p}$, is it true that $f(z.w) \le f(z)+f(w)$?

Let $p \in \Bbb R$ with $p>1$ and $f_p : \Bbb C_{\ne 0} \rightarrow \Bbb R_{\ge 0}$ defined by $$f_p(z)=\frac{|z-1|}{(|z|^p+1)^{1/p}}$$ I'm trying to prove that $$f_p(z.w)\le f_p(z)+f_p(w)$$ I ...
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2answers
46 views

Are all index 2 subgroups of $(\mathbb{Z} /2\mathbb{Z})^n$ isomorphic to $(\mathbb{Z} /2\mathbb{Z})^{n-1}$

In particular, consider the homomorphism from $(\mathbb{Z} /2\mathbb{Z})^n \to {\pm1}$ sending $\{ \epsilon_i \}^n$ to $\prod \epsilon_i$ where $\epsilon_i = \pm1$. The kernal of this homomorphism is ...
2
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2answers
47 views

Can a nonabelian group $G$ have a normal abelian subgroup $H$ with $[G:H]=3$?

If $H$ is a cyclic group, for instance, and $[G:H]=2$, then $G$ is a dihedral group with elements equal to the possible states of a regular polygon upon which rotations and reflections are applied. In ...
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1answer
36 views

Find the amount of subgroups of order $3$ and $21$ in non-cyclic abelian group of order $63$

Find the amount of subgroups of order $3$ and $21$ in non-cyclic abelian group of order $63$. In first case I found the amount of elements that have order $3$ - there are $8$ of them, in second case ...
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1answer
58 views

For any finite abelian group $G$, there is an integer $m$ with $G$ isomorphic to a subgroup of $U(\mathbb{Z}_{m})$.

I want to prove if the following assertion from Rotmans Advanced Algebra page 205 is true: For any finite abelian group $G$, there is some integer $m$ with $G$ isomorphic to a subgroup of $U(\mathbb{...
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1answer
41 views

What is this abelian group notation?

While studying abelian groups, I came across the abelian group $$G = \frac{1}{4} \mathbb{Z} / \mathbb{Z}$$ what is this group? I've never seen this notation before?
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votes
1answer
77 views

The set of bilinear forms is a right $(R \otimes R)$-module

Let $V$ and $A$ be abelian groups. An $A$-valued bilinear form on $V$ is a $\mathbb{Z}$-module homomorphism $$\beta : V \otimes_{\mathbb{Z}} V \rightarrow A$$ Now, let $V$ be a left $R$-module, where $...
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1answer
13 views

Example of a bilinear form of abelian groups

Let $X$ and $Y$ be abelian groups. Then, a $Y$-valued bilinear form on $X$ is a $\mathbb{Z}$-module homomorphism $$\alpha: X \otimes_{\mathbb{Z}} X \rightarrow Y$$ How does this relate to the standard ...
0
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1answer
55 views

Normal group of product of abelian and non-abelian simple groups

I am trying to understand the structure of normal groups of $G=A\times H$ where $A$ is an abelian group and $H$ is a direct product of non-abelian simple groups (both are finite). I want to show that ...
5
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2answers
231 views

Does every subgroup of an abelian group have to be abelian?

My original problem is to show that E/L is an abelian extension over L and L/F is an abelian extension over F, given that E/F is an abelian extension over F and that L is a normal extension of F such ...
6
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1answer
77 views

How do basic results in representation theory change going to positive characteristic fields and/or not algebraically closed?

I'm working on a project linking graph theory, model theory and representation theory, and am interested in how some results change if we work over fields of positive characteristic or fields that are ...
6
votes
1answer
91 views

Number of subgroups of an abelian p-group

Let $p$ be a prime number and let $n\in \mathbb{N}$. I know that every abelian group of order $p^n$ is uniquely a direct sum of cyclic groups of order $p^{\alpha_i}$ where $\sum \alpha_i = n$. Now the ...
2
votes
1answer
74 views

Let $S=\{x~|~x\in G$ and $x^2 \in H\}$. Show that $S$ is a subgroup of $G$ for $H<G$, $G$ abelian.

The question states: Let $G$ be an Abelian group with subgroup $H < G$. Let $S=\{x~|~x\in G$ and $x^2 \in H\}$. Show that $S$ is a subgroup of $G$. My proof is different than what is in the ...
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vote
3answers
56 views

Suppose that $H_1/H_2$ is Abelian. Show that $H_1 N / H_2 N$ is Abelian.

Suppose $G$ is a group and $H_1$, $H_2$, $N$ are subgroups of $G$. $N$ is a normal subgroup of $G$ and $H_2$ is a normal subgroup of $H_1$. Suppose that $H_1/H_2$ is Abelian. Show that $H_1N/H_2N$ is ...
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1answer
20 views

Enough injectives in the category of torsion abelian groups

Claim: The category of torsion abelian groups has enough injectives. I thought I had a proof of this, but discovered a mistake in my proof. I was trying to use the techniques of the usual proof that ...
1
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1answer
43 views

Number of generators of subgroup

I am trying to prove the following. let $G$ be a finitely generated abelian group, and $H<G$ a subgroup such that there exists a subgroup $K<G$ and we can write $G=H \oplus K$. Is it true that ...
1
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1answer
38 views

Direct sum of $n$ (infinite) cyclic groups isomorphic to direct sum of $n$ copies of $\mathbb{Z}$?

I'm currently selfstudying some algebra and i am currently covering the various equivalent definitions of free abelian groups. However, in order to understand why these definitions are indeed ...
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1answer
24 views

Product of elements from Direct summand in the direct sum abelian group

Let $G = H \oplus K$ be abelian group. Now, I follow the definition of direct sum from Wikipedia which is Now, we choose two element (both of them are not identity) $h \in H$ and $k \in K$ ($h, k \...
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0answers
44 views

Hungerford Chapter 2 Section 2 Problem 2 WITHOUT using the structure theorem of finite abelian groups

Let $G$ be a finite abelian group and $x$ an element of maximal order. Show that$\langle x \rangle$is a direct summand of $G$. Use this to obtain another proof of Theorem 2.1. Theorem 2.1: Every ...
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1answer
35 views

Group is soluble if and only if quotient is abelian

So a group G is soluble if and only if it has a subnormal series $$ \{ 1\} =G_0 \ \triangleleft \ G_1 \ \triangleleft \ ... \ \triangleleft \ G_n=G $$ where all quotient groups $G_{i+1}/G_i $ are ...
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2answers
37 views

How to find number of abelian subgroups of diheral group? [closed]

How to find number of abelian subgroups of diheral group $D_n $? Attempt: I have counter-examples for $n=1,2$ so I know that it isn't true for $n<3$. Is it true for $n\ge 3$? How do you know this?...
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2answers
81 views

Mapping from $\mathbb Z/2\mathbb Z \times \mathbb Z/3\mathbb Z \times \dots\times \mathbb Z/p_n\mathbb Z$ to $\mathbb Z/p_n\#\mathbb Z$.

I know $\mathbb Z/2\mathbb Z \times \mathbb Z/3\mathbb Z \times \dots\times \mathbb Z/p_n\mathbb Z$ is isomorphic $\mathbb Z/p_n\#\mathbb Z$ (where $p_n\#$ is the primorial of primes up to $p_n$) by ...
3
votes
1answer
57 views

Find $H<G$ so that $\{(x, y) | xx^{−1} y^{−1} \in H\}$ is not an equivalence relation on $G$.

The question is as follows: Find an example of a group $G$ with a subgroup $H$ so that $$\{(x, y) | xx^{−1} y^{−1} \in H\}$$ is not an equivalence relation on $G$. I've just been working on this ...
2
votes
3answers
120 views

What group is $\langle (1,1,1) \rangle$ in $\mathbb{Z}_5 \times \mathbb{Z}_4 \times \mathbb{Z}_8$?

I know that $\langle (1,1,1) \rangle$ in $\mathbb{Z}_5 \times \mathbb{Z}_4 \times \mathbb{Z}_8$ is isomorphic to $\mathbb{Z}_{40}$, but is there a way of writing what group it is (not what it's ...
1
vote
1answer
63 views

Power automorphism of elemantary abelian group

I proved that a subgroup A normalizes every subgroup of the minimal normal subgroup $N$, $N$ is an elementary abelian group of order $p^n$, $n>1$. It is clear that A induces a power automorphism ...
0
votes
1answer
82 views

Let $G= \langle a \rangle$ be a cyclic group and $H$ a subgroup of $G$. Show $(G/H,*)$ is a cyclic group with generator $aH$.

Let $G=\langle a \rangle$ be a cyclic group and $H$ a subgroup of $G$. Show $(G/H,*)$ is a cyclic group with generator $aH$. Also find a group $K$ with normal subgroup $L$ such that $(K/L,*)$ is ...
0
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0answers
17 views

Subgroup of ${\rm Aut}\,(\widehat{\mathbb{Z}})$

Ribes and Zalesskii Corollary 4.4.8 show that the group of continuous automorphisms of $\widehat{\mathbb{Z}}$ satisfies ${\rm Aut}\,(\widehat{\mathbb{Z}})\cong\mathbb{Z}_2\times\frac{\mathbb{Z}}{2\...
3
votes
1answer
54 views

Number System with Torsion

Introduction: A number system, long known but seldom seen, is (re)introduced for which some elements are torsion and some are torsion-free. A topology question and an analytic number theory question ...
0
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2answers
34 views

$|G:H|=2$ and $H$ abelian then $H \subset Z(G)$

I have to show whether this is true or false: $|G:H|=2$ and $H$ abelian then $H \subset Z(G)$ I have proved that $H \triangleleft G$, but with this I can show that if $h \in H, g\in G$ then exists ...
1
vote
1answer
142 views

Prove that $\Bbb Z\oplus \Bbb Z_2$ is not isomorphic to $\Bbb Z∗\Bbb Z_2$.

I need to prove that $\Bbb Z\oplus \Bbb Z_2$ is not homeomorphic to the free product $\Bbb Z∗\Bbb Z_2$. I know that $\Bbb Z\oplus \Bbb Z_2$ is abelian while the free product $\Bbb Z∗\Bbb Z_2$ is not ...
5
votes
1answer
104 views

Does the Burnside $\mathbb Q$-algebra $A$ of a group depend only on $\dim_{\mathbb Q}A$?

The Burnside $\mathbb Q$-algebra $\mathbb QB(G)$ of a group $G$ is usually considered only when $G$ is finite; see Section 3.1 of the text [1] Serge Bouc, https://pdfs.semanticscholar.org/aff3/...
3
votes
1answer
95 views

Prüfer Groups and Product Topologies

For each $p\in\mathbb{P}$, the Prüfer group $\mathbb{Z}(p^{\infty})$ is a divisible abelian group which can be given the discrete topology or the topology it inherits via its identification as a ...
1
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0answers
24 views

Lattice and abelian Lie groups

Let $\Lambda$ be a discrete (lattice) subgroup of $\mathbb R^n$. Let $V:=\langle \Lambda\rangle_\mathbb R$. Define the abelian Lie group $G:=V/\Lambda$. Now if $H$ is a Lie subgroup of $G$. Does ...
-1
votes
1answer
74 views

What are the elements of $\Bbb Z\oplus \Bbb Z_2$ and $\Bbb Z∗\Bbb Z_2?$ [closed]

What are the elements of the following groups? (1) $\Bbb Z\oplus \Bbb Z_2$ , (2) $\Bbb Z∗\Bbb Z_2$ Thanks in advance.
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0answers
35 views

What is a simply presented group?

I have some background in commutative ring theory. At the moment I am going through factorization theory of integral domains. I found out that it is a conjecture, that every Abelian group is the ...