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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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13 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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0answers
25 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
36 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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0answers
15 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
10 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 ...
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1answer
36 views

Let $k$ be the largest order of elements of finite abelian group A. Prove, that the order of any element in $A$ divides $k$. [on hold]

Let $k$ be the largest order of elements of finite abelian group A. Prove, that the order of any element in $A$ divides $k$. Finite abelian group A is isomorphic to $\mathbb{Z}_{p_{1}^{t_{1}}} \...
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1answer
48 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 ...
4
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2answers
193 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
70 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
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1answer
88 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 ...
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1answer
61 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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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
17 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
41 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 ...
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1answer
35 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
22 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
40 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
42 views

Abelian Groups and order [closed]

Find the least value of 𝑛 such that there are six non-isomorphic Abelian groups of order 𝑛, and justify your argument.
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1answer
30 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
36 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
68 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 ...
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1answer
56 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 ...
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3answers
110 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 ...
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0answers
20 views

The underlying group of a positive-dimensional vector space is not free abelian

This is homework: "prove that the underlying group $G$ of a positive-dimensional vector space is not free abelian". My solution is the following. If the field of definition has positive ...
1
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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 ...
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1answer
65 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 ...
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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
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1answer
52 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 ...
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2answers
32 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 ...
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1answer
114 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 ...
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1answer
94 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/...
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1answer
94 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 ...
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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 ...
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1answer
66 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 ...
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0answers
38 views

A subgroup of direct product of countable copies $\mathbb{Z}$

This is an example from Rotman's Homological Algebra (p.122, old edition). Let $G$ be the direct product of countable copies of $\mathbb{Z}$ (say, group of sequences of integers). Fix a prime $p$. ...
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0answers
23 views

Structure of l-Sylow group

I have Z_n (the cyclic group of order n) and I'll let p, q, r, be three different prime numbers and I'll consider the abelian group: A= Z_p^2 * Z_(p*q^2) * Z_(q^2 *r) * Z_r of order p^3 * q^4 * r^2. ...
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2answers
82 views

Give an example of a group $G$ such that $G/Z(G)$ is not abelian. [closed]

Give an example of a group $G$ such that $G/Z(G)$ is not abelian. I am having trouble understanding what the group $G/Z(G)$ looks like. Because of that I am having troubles answering the above ...
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1answer
46 views

finiteness of an abelian topological group

Let A be an abelian (Hausdorff) topological group. Assume that (1) the set of its torsion elements, and (2) a finitely generated subgroup are dense subsets of A. My question: must A be finite? (...
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1answer
160 views

Is there a surjective morphism from an infinite direct product of copies of $\mathbb{Z}$ to an infinite direct sum of copies of $\mathbb{Z}$?

Is there a surjective morphism $\mathbb{Z}^I\to \mathbb{Z}^{(J)}$ for some $I,J$? i) I'm asking about group morphisms ii) $\mathbb{Z}^{(J)}$ denotes the direct sum of $J$ copies of $\mathbb Z$ iii) ...
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1answer
43 views

$f: G \to C^{×}$ is homomorphism. |G|=n is abelian .prove that $\sum_{g \in G} f(g)$ is 0 or n [duplicate]

Let $$f: G \to C^{×}$$ is homomorphism, whereas $C^{×}$ is the multiplicative group of non-zero complex numbers, |G|=n and G is abelian. Prove that \begin{align} \sum_{g \in G}f(g)&=0,\...
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1answer
46 views

In a group $G$, if $a^3=e$ for all $a$ belongs to $G$, then is $G$ abelian? [duplicate]

It's easy to show that $G$ is abelian if $a^2=e.$ Can't seem to figure out how to prove/disprove this.
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1answer
35 views

Abelian $p$-group and proof of the existence of cyclic subgroup

Consider the theorem Let $G$ be a finite Abelian group with order $|G|=p^n$ and $a$ an element of maximal order in $G$, then there is a subgroup $H$ of $G$ such that $G\cong |a|\times H$. I'm ...
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1answer
33 views

Homomorphism on Abelian Group

We are given that $G$ is an abelian group of order $n.$ If $f: G \rightarrow \mathbb{C}^*$ is any homomorphism, then show that $\sum_{g \in G} |f(g)| = n$ Please give a hint rather than the answer ...
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1answer
78 views

Alternative proof of the Fundamental Theorem of Abelian Groups??

While the literature has many variations, all the published proofs I know induct on the number of generators. Thus they start will an abelian group $A$ and build up a direct sum of cyclic groups ...
2
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1answer
99 views

Non-abelian Groups of Order $p^3$

I am trying to show that a non-abelian group $G$ of order $p^3$ is isomorphic to one of two groups constructed on page 48 of these group theory notes (see examples 3.14 and 3.15 on that page). Here ...
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2answers
35 views

What is the difference between Cauchy's Theorem and Cauchy's Theorem for Abelian Groups?

These theorems seem to be identical but for some reason, the requirement that a group is finite AND abelian is sometimes stated instead of just finite. Could someone let me know if there is a ...
4
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2answers
52 views

Showing there is a unique group table for $\{1, a,b,c\}$ such that there is no element of order $4$. [duplicate]

Assume $G = \{1, a,b,c\}$ is a group of order $4$ with identity $1.$ Assume also that $G$ has no elements of order $4$. Show that there is a unique group table for $G$. Also show that $G$ is abelian. ...
4
votes
2answers
61 views

Infinite group with the order of abelian subgroup bounded

In Isaac's Finite Group Theory Page 28, it states: There exist infinite groups in which the abelian subgroups have bounded order. I fail to construct such group. In fact, I'm only able to deduce ...
2
votes
1answer
44 views

Abelian groups about rationals

Is the set $\mathbb{Q}$ under $×$ an abelian group? It is sure for $\mathbb{Q} - {0}$, but i think the whole set of rationals is not an abelian group as $0 × a = a × 0 = 0$, but the identity element ...