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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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0answers
18 views

Hungerford Chapter 2 Section 2 Problem 2 about the structure of finite abelian groups [duplicate]

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
30 views

Abelian Groups and order [on hold]

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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2answers
66 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 ...
5
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3answers
552 views

A simple question on characteristic subgroups

Suppose $P$ is a finite abelian $2$-group and $U$,$V$ are characteristic subgroups of $P$ such that $|V:U|=2$. Does it follow that $P$ has a characteristic subgroup of order $2$?
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1answer
93 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
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? [on hold]

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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3answers
105 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 ...
3
votes
1answer
55 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 ...
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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0answers
18 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 ...
2
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4answers
199 views

“Classify $\mathbb{Z}_5 \times \mathbb{Z}_4 \times \mathbb{Z}_8 / \langle(1,1,1)\rangle$”

I have a question that says this: Classify $\mathbb{Z}_5 \times \mathbb{Z}_4 \times \mathbb{Z}_8 / \langle(1,1,1)\rangle$ according to the fundamental theorem of finitely generated abelian groups. ...
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2answers
154 views

In what locally compact abelian groups does $\mathbb{Q}$ embed densely?

I know that there is classification of local fields, but here is a closely related question: Can the additive group of $\mathbb{Q}$ be a proper dense subgroup of a locally compact abelian group, whose ...
2
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1answer
979 views

Let $G,H$ groups, $H$ abelian, prove $\def\Hom{\operatorname{Hom}}\Hom(G,H)$, with the entrywise sum $(f+g)(x)=f(x)+g(x)$ is an abelian group.

I'm working on a proof to show that for a group $G$ and an abelian group $H$, the set of all homomorphisms $\def\Hom{\operatorname{Hom}}\Hom(G,H)$ from $G$ to $H$ is an abelian group. I just want to ...
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7answers
14k views

Give an example of a noncyclic Abelian group all of whose proper subgroups are cyclic.

I've tried but I could not find a noncyclic Abelian group all of whose proper subgroups are cyclic. please help me.
3
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1answer
93 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 ...
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 ...
0
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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
19 views

Abelian homomorphism inverse proof [closed]

I have an abelian group $G$ and a mapping $m:G\to G$ where $m(a)=a^{-1}\forall a\in G$. Is it possible to prove that $m$ is a homomorphism?
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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\...
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1answer
65 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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1answer
113 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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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 ...
15
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4answers
14k views

Proof that all abelian simple groups are cyclic groups of prime order

Just wanted some feedback to ensure I did not make any mistakes with this proof. Thanks! Since $G$ is abelian, every subgroup is normal. Since $G$ is simple, the only subgroups of $G$ are $1$ and $G$,...
1
vote
1answer
63 views

abelian group, direct summand

Let $G$ be an abelian group, and $H$ and $N$ is a subgroup. And assume $$ G / H \cong N \ . $$ Are these assumptions sufficient to show that $G$ is direct sum of H and N? Thank you for your help!
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 ...
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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
36 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 $p$. ...
2
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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 ...
1
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1answer
112 views

$p$-primary part of an abelian group

Let $M$ be a torsion abelian group. For a prime $p$, let $\Bbb Z_p$ be the $p$-adic integers (seen as an abelian group), and $M[p^{\infty}] := \{m \in M : p^n m= 0 \text{ for some } n \geq 0\}.$ Is ...
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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. ...
4
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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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2answers
81 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 ...
12
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1answer
154 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) ...
2
votes
1answer
75 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 ...
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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,\...
0
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1answer
34 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 ...
0
votes
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
717 views

Prove that if a and b are elements of an Abelian group G, with o(a) = m and o(b) = n, then (ab)^{mn}

(a) Prove that if a and b are elements of an Abelian group G, with o(a) = m and o(b) = n,then (ab)^{mn} = e. Indicate where you use the condition that G is Abelian. (b) With G, a, and b as in part (a)...
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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
830 views

abelian group as Z module

How Would you prove that every abelian group can be understood as a Z-Module in a unique way? I would guess that you would have to prove its bijective, but not sure how to go about this
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4answers
12k views

Need to prove that $(S,\cdot)$ defined by the binary operation $a\cdot b = a+b+ab$ is an abelian group on $S = \Bbb R \setminus \{-1\}$.

So basically this proof centers around proving that (S,*) is a group, as it's quite easy to see that it's abelian as both addition and multiplication are commutative. My issue is finding an identity ...
8
votes
1answer
663 views

Group $\mathbb Q^*$ as direct product/sum

Is the group $\mathbb Q^*$ (rationals without $0$ under multiplication) a direct product or a direct sum of nontrivial subgroups? My thoughts: Consider subgroups $\langle p\rangle=\{p^k\mid k\in \...
4
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2answers
51 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. ...
2
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1answer
96 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 ...
4
votes
2answers
60 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 ...
0
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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 ...
3
votes
2answers
80 views

Why does $a^{m_1}=a^{m_2}$ imply $a^{m_1-m_2}=e$?

I was reading this answer. I understand almost all of it. However, there is still one thing that continues to puzzle me. How should I prove for sure that, in this example, if $m_1\neq m_2$ and $a^{...
4
votes
1answer
60 views

Commutative subtraction

It is well known that subtraction is not commutative in general. However, it is commutative in some groups: $\mathbb I$, $\mathbb C_2$, $\mathbb K_4$. I am trying to understand the logic. ...
1
vote
1answer
27 views

Direct product decomposition of the group of complex roots of unity

I'm studying $p$-adic numbers (Robert's "A course in $p$-adic analysis) and, at page 41, the author states that, for every prime $p$, the group $\mu$ of all complex roots of unity has a direct product ...