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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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21 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
32 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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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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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 ...
3
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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 ...
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3answers
106 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
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 ...
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
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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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
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 ...
5
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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/...
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 ...
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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
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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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 $p$. ...
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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
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 ...
4
votes
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? (...
12
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1answer
155 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) ...
-1
votes
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
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 ...
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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 ...
2
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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 ...
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 ...
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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
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. ...
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 ...
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
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 ...
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0answers
32 views

Abelian subgroups of a simple group

Let $G$ be a finite non-abelian simple group, and $A \leqslant G$ be an abelian subgroup. How large $A$ can be? There exists any bound of the type $|A| \leq |G|^r$ for some $r<1$? How can I prove ...
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. ...
3
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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^{...
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1answer
28 views

Find the isomorphism type

Consider the abelian group $G$ generated by $a$, $b$ and $c$ and determined by the following relations \begin{aligned} 3 a+9 b+9 c &=0 \\-3 b+9 c &=0 \end{aligned} determine the isomorphism ...
5
votes
1answer
110 views

G is a group and $(ab)^3=a^3b^3$ for all $a,b \in G$. Prove (or disprove with a counterexample) that if $(ab)^3=(ba)^3$, then $ab=ba$.

Proposition. Let $G$ be a group such that $(ab)^3=a^3b^3$ for all $a,b \in G$. If $(ab)^3=(ba)^3$, then $ab=ba$. Is it true or false? So far I've only been able to prove that powers of $a$ commute ...
1
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2answers
41 views

prove that for $G$ a multiplicative nonabelian group of order $pq$, where $p$ and $q$ are prime numbers, any proper subgroup of $G$ is abelian

I need to prove that for $G$ a multiplicative nonabelian group of order $pq$, where $p$ and $q$ are prime numbers, any proper subgroup of $G$ is abelian. I use that, from Lagrange's theorem, the order ...
2
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1answer
21 views

Relationship between Archimedean and Divisible ordered groups

Let $(G,+,\leq)$ be a linearly ordered abelian group (i.e. the order is total and compatible with the sum) and $n\cdot x$ denote the classical action of $\mathbb{Z}$ over $G$ (i.e. $0$ for $n=0$, sum ...
2
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0answers
31 views

Modifying long exact sequences

Let $$\dots A_i\stackrel {f_i}\to B_i \stackrel {g_i}\to C_i \stackrel {h_i}\to A_{i+1}\to \dots$$ be a long exact sequence of Abelian groups. Is it true that if there are maps $k_i:D_i\to E_i$ such ...
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0answers
14 views

All the $m$th powers commute with each other and all the $n$th powers commute with each other, $m$ and $n$ relative prime, is abelian. [duplicate]

Show that a group in which all the $m$th powers commute with each other and all the $n$th powers commute with each other ,$m$ and $n$ relatively prime, is abelian. How to do this ? we know $1=mx+yn$ ...
6
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2answers
85 views

Associocommutativity

One thing I've noticed is that addition and multiplication both form commutative groups over the reals, but subtraction, division, and exponentiation are neither associative nor commutative. Ignoring ...
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0answers
33 views

Group Homomorphisms into an Abelian Group

The following comes from Hungerford's Algebra. [Prove that if] $f: G \to H$ is a homomorphism, $H$ is abelian and $N$ is a subgroup of $G$ containing $\ker f$, then $N$ is normal in $G.$ A ...
3
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1answer
31 views

if $G$ is a divisible group then any subgroup of $G$ is also divisible

Is it true that if $G$ is a divisible group then any subgroup of G is also divisible? I know that if $H \leq G$ then for any $h \in H$ and for any $k \in \mathbb{N}$ then there exists $x \in G$ such ...
5
votes
1answer
116 views

finite abelian groups tensor product. [closed]

Is the following question obvious ? Let $G$ be an abelian group, such that for any finite abelian group $A$, we have $G\otimes_{\mathbf{Z}}A=0$, does it mean that $G$ is a $\mathbf{Q}$-vector space ?...
0
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0answers
13 views

Let be $G$ a group if order $p^n$ where p is a prime and n a natural number. Prove that there exist normal subgroups… [duplicate]

Let be $G$ a group of order $p^n$ where $p$ is a prime and $n$ a natural number. Prove that there exist normal subgroups $N_{1}, N_{2}, ..., N_{n}$ of $G$ such that $N_{1} < N_{2}<...<N_{n}$ ...