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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Folner sequence of $\mathbb{Z}[\frac{1}{2}]$

Consider $\mathbb{Z}[\frac{1}{2}]$ consisting of rational numbers of the form $k2^l$ with $k,l\in\mathbb{Z}$. Under addition and discrete topology $\mathbb{Z}[\frac{1}{2}]$ is a discrete abelian ...
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1answer
26 views

Abelian-by-(finite abelian) [on hold]

hope you all doing fine. I have a question. Is it true that a abelian-by-(finite abelian) group is also (finite abelian)-by-abelian? Thanks.
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22 views

Finitely generated abelian groups and finite index subgroups

I want a proof or reference for the following fact: "Let $G$ be a finitely generated abelian group and let $\phi:G\to G$ be an injective homomorphism. Then the index $[G:\phi(G)]$ is finite." I ...
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1answer
38 views

About a proof regarding a property of groups of order $pq$ where $p$ and $q$ are primes

I'm studying right now Automorphisms in Dummit & Foote's Abstract Algebra (Section 4.4). In pages 135-136, the following example is given: and here's Proposition 16 muntionned in the ...
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1answer
30 views

A questions on the groups by a copy of $\Bbb Z$

Let $G$ be an abelian group and $H$ a subgroup of $G$ such that $G/H$ contains a copy of $\Bbb Z$. Is this true that $G$ contains a copy of $\Bbb Z$? ($\Bbb Z$ is the group of integer numbers)
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1answer
34 views

If $G_3$ is finitely generated abelian group then there is a short exact sequence with $G_2$ and $G_1$ free groups?

Let $G_i$ be abelian Groups. A exact sequence of the form $ 0 \to G_1 \to G_2 \to G_3 \to 0$ is called a short exact sequence. Is the following statement true? If $G_3$ is finitely generated then ...
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1answer
79 views

Fundamental Theorem of Abelian Groups

From fundamental theorem of finite abelian groups I can say any finite abelian group $G$ is isomorphic to direct sum of cyclic groups i.e, $G\cong Z_{{p_1}^{i_1}}\oplus Z_{{p_2}^{i_2}}\oplus ...
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1answer
51 views

Abelian group and their subgroups

Is it true that If an abelian group has subgroups of order m and n respectively then it has a subgroup whose order is the least common multiple of m and n? If it is then can anyone explain it with a ...
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42 views

Generating Sets for Subgroups of $(\Bbb Z^n,+)$.

The question Finite Generated Abelian Torsion Free Group is a Free Abelian Group led me to conjecture and prove an interesting thing about generating sets for $\Bbb Z^n$ and certain subgroups. If ...
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0answers
36 views

Finding a property for $G/Z(G)$ where $G$ is a nonabelian group

If $G$ is non-abelian group and $Z(G)$ is it's center, what is the least property for $G$ such that $\frac{G}{Z(G)}$ is abelian?
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4 views

Is there a characterization for discrete abelian torsion-free and reduced groups?

Is there a characterization for discrete abelian torsion-free and reduced groups? A group is called reduced if it contains no nontrivial divisible subgroups.
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1answer
32 views

discrete subgroups of euclidean space

I'm trying to prove this proposition: http://groupprops.subwiki.org/wiki/Every_discrete_subgroup_of_Euclidean_space_is_free_Abelian_on_a_linearly_independent_set That every discrete subgroup of ...
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2answers
52 views

If $f\otimes_\mathbb{Z}\mathbb{Z}/(p)\colon M\otimes_{\mathbb{Z}}\mathbb{Z}/(p)\to N\otimes_\mathbb{Z} \mathbb{Z}/(p)$ is onto for all $p$, $f$ onto?

This lemma is used in a theorem I'm reading, with no proof. Suppose $f\colon M\to N$ is a morphism of free, finitely generated $\mathbb{Z}$-modules. Then if $f\otimes_\mathbb{Z}\mathbb{Z}/(p)$ is ...
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1answer
24 views

setting abelian group in non-abelian group

Is it right to say for every (finite) abelian group $H$ there is non-abelian group $G$ such that $Z(G)=H$, where $Z(G)$ is the center of $G$?
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1answer
29 views

A characterization of an abelian group

Let $G$ be an abelian group. Is there a characterization of $G$ whenever every subgroup of $G$ is a direct summand of $G$?
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1answer
74 views

A group is abelian if and only if the center of the group is all the group

Isn't it the same to say that a group is abelian, and that the center of the group is all the group? I have an exercise to prove that this is true, and it's exactly one stroke for each direction of ...
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2answers
60 views

What is the rank of $\mathbb{Q}$ over $\mathbb{Z}$?

What is $\operatorname{rank}_{\mathbb{Z}} \mathbb{Q}$? I think it is $\aleph_0$, but cannot figure out how the basis would look like. Thank you in advance. (I faced this when proving ...
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1answer
14 views

$A/A^{p}\cong A_{p}$ for finite abelian (additive) gp. and prime $p$.

Let $A$ be a finite abelian (additive) gp. and $p$ be a prime. I want to show $A/A^{p}\cong A_{p}$ where $A^{p}:=\left\{pa:a\in A\right\}$ and $A_{p}:=\left\{a\in A:pa=0\right\}$.(I want to show ...
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0answers
17 views

What are the character functions of $\mathbb{Z}_N \times \mathbb{Z}_N$ ?

$\mathbb{Z}_N \times \mathbb{Z}_N$ is an Abelian group which I can think of to consist of all tuples of the form $(\omega ^a, \omega^b)$ where $0 \leq a,b \leq (N-1)$ and $\omega = e^{ \frac{2 \pi i ...
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2answers
110 views

How to prove the group $G$ is abelian?

Question: Assume $G$ is a group of order $pq$, where $p$ and $q$ are primes (not necessarily distinct) with $p\leqslant q$. If $p$ does not divide $q-1$, then $G$ is Abelian. I know that if the ...
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1answer
41 views

Cubic Planar Graphs have $2^m-1$ Hamilton Cycles

I looked at the symmetric difference of hamilton cycle (HC) in cubic planar graphs and found that, together with the empty graph, they build a subgroup of the abelian group $\Omega$ of symmetric ...
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1answer
26 views

A question on finite abelian groups

Let $m_1,\dots,m_k$ be positive integers. Are there positive integers $d_1,\dots,d_k$ such that $d_i|d_{i+1}$ and $$ \oplus_{i=1}^k \mathbb{Z}/m_i\mathbb{Z}\cong \oplus_{i=1}^k ...
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1answer
21 views

Definition of quasi-cyclic and full rational groups

In Unit Groups of Classical Rings by Karpilovsky, p.96, we can see this theorem: Let $G$ be a divisible abelian group. Then $G$ is a direct product quasi-cyclic and full rational groups. I want ...
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1answer
31 views

Exsitence of element of a certain order in an infinite abelian group

I came up with the following question reading this(Finite Abelian Groups question). Let $G$ be an abelian group. Suppose there is an integer $n \ge 1$ such that $nG = 0$. Let $m$ be the smallest ...
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1answer
33 views

Quotient group of free groups

Let $G=\langle g_1,\ldots,g_k\rangle$ be a free abelian group generated with $g_1,\ldots,g_k$ and let $H=\langle g_{r+1},\ldots,g_k\rangle$ be a free abelian subgroup of $G$. Is it then the case that ...
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1answer
45 views

Given $K(\alpha)/K$ and $K(\beta)/K$ abelian extensions, prove that $K(\alpha + \beta)/K$ is an abelian extension.

Problem: Let $K(\alpha)/K$ and $K(\beta)/K$ algebraic field extensions so that their respective Galois groups are abelian. Prove that the Galois group of the field extension $K(\alpha + ...
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4answers
127 views

Prove that $\mathbb Z^n$ is not isomorphic to $\mathbb Z^m$ for $m\neq n$

Prove that groups $\mathbb Z^n$ is not isomorphic to $\mathbb Z^m$ for $m\neq n$ My try: Let $\mathbb Z^n\cong \mathbb Z^m $ .To show that $m=n$. Case 1:Let $m>n$.Now that $\mathbb Z^m$ has $m$ ...
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1answer
68 views

Prove G is Abelian

Let $G$ be a group and $a,b\in G$. given that $(ab)^k=a^k b^k$ and $(ab)^{k+2}=a^{k+2} b^{k+2}$ for some $k\in \mathbb N$. prove that $G$ is abelian. So far my attempt was: ...
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1answer
51 views

Matrix and Abelian groups question

Let $A$ be a Matrix: $$ A=\begin{pmatrix} 1 & 2\\ 4 & 1 \end{pmatrix} $$ Let $f\colon v\to Av$ be a homomorphism from $Z^2$ to $Z^2$. Find a base $(v_1,v_2)$ to $Z^2$ and $2$ integers ...
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1answer
47 views

characterisation of direct sum of abelian groups

The direct sum of abelian groups can be defined in several equivalent ways, but I have some problem proving the equivalence. Definition 1: Given abelian groups $A$ and $B$, the direct sum is defined ...
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4answers
114 views

Representation of an abelian group

Without using the structure theorem, how do I prove b? I struggle with the proof of injectivity. Any tips? Problem: Let $G$ be a finite Abelian group. (a) Prove that the group homomorphisms $\chi ...
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102 views

Finite Abelian Groups question [closed]

Let A be a finite abelian group. Let m be the smallest natural number such that ma=0 for every a in A. Prove that there is an element in A of the order m
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Which abelian groups have only a single composition series?

Cyclic groups of composite powers don't: for example, $1=C_1\triangleleft C_3\triangleleft C_6 $ and $1=C_1\triangleleft C_2\triangleleft C_6 $ are both composition series for $C_6$. But cyclic ...
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1answer
73 views

How can I prove that this Group is Abelian? [duplicate]

$(G,\cdot)$ a group. If $\exists n\in \mathbb{Z} $ such that $(a\cdot b)^{n+i}=a^{n+i}\cdot b^{n+i}$ for $i=0,1,2.$ $\forall a,b \in G$. Prove that $(G,\cdot)$ is Abelian. I'm not sure how to prove ...
3
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1answer
27 views

Describing all elements in a factor group using the structure theorem for finitely generated abelian groups

On an exam I took a couple of weeks ago there was this question, which I would like to review as I did not figure it out during the exam. We were given the matrix $$A = \begin{bmatrix} 1 & 2 & ...
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1answer
37 views

Reference Request: Subgroup of free abelian group is free abelian

I have the following reference Question, meaning that I search for a reference for the following statement: Let $F$ be a free abelian group of finite rank and $U$ be a subgroup. Then there is a basis ...
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40 views

Direct sum notation

I was reading the direct sum of the groups and the index notation looks little bit strange for me. Group $G = \bigoplus_{\alpha < \beta}\mathbb Zx_{\alpha},$ where $\beta$ be an ordinal. Is this ...
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1answer
43 views

Subgroup proof verification.

Let $G$ be an abelian group, K is a fixed positive integer. $H$={$a\in$ $G$ $|$ $|a|$ divides K} . Prove that $H$ is a subgroup of $G$. My way of proving (Let me know how I could make it better or ...
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2answers
50 views

Finitely generated abelian group with certain properties

Problem Characterize all finitely generated abelian $G$ such that every proper subgroup of $G$ is cyclic, $G$ contains exactly two proper subgroups, and for each pair of subgroups $S$,$T$ in $G$ ...
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3answers
67 views

Prove that $G=\{z \in \mathbb{C}: |z|=1\}$ is an abelian group [closed]

Prove that $G=\{z \in \mathbb{C}: |z|=1\}$ is an Abelian group with the multiplication operation of complex numbers. In other words, we want to prove that: $$\begin{align} &(G_1).\; ...
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10 views

A representative subspace for the cosets of another subspace $H$.

Suppose I have a subspace $H \leq V$ over a commutative ring (or if I can't get what I want with that generality, a field). I would like to specify a subspace $K$ such that: -$K \cap H = \{ \vec{0} ...
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1answer
70 views

Structure of the unit group $(\mathbb{Z}[i]/8\mathbb{Z}[i])^\times$

I know that $\mathbb{Z}[i]/8\mathbb{Z}[i]=\{a+ib \mid a,b\in\mathbb{Z}_8\}$. But I'm not able to comprehend what $(\mathbb{Z}[i]/8\mathbb{Z}[i])^\times$ is. Can someone please help me get its ...
3
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0answers
39 views

Nontrivial homomorphisms from G to T

Let $G$ be a compact metric abelian group. $T$ be the circle group. Let $\mathcal{A}$ be the set of all finite linear combinations of continuous homomorphisms from $G \to T$. I want to show ...
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2answers
15 views

Conjugate closure and factor group

Let $N\unlhd K$ be a normal subgroup of a given group $K$ and let $$q:K\to K/N$$ be the natural quotient map. Let $A\subseteq K$ be a subset of $K$ and let the conjugate closure of $A$ in $K$ be ...
3
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1answer
51 views

Prove the group is a direct product [closed]

Let $G$ be an abelian group of finite order $n = mk$ with gcd$(m,k) = 1$. For $r=m,k$, let $G(r) = \{g \in G: g^r = 1 \}$ . Prove that $G = G(m) \times G(k)$.
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1answer
65 views

Showing that the product $x*y := \frac{x+y}{xy+1}$ is a group operation on $(-1, 1)$ [duplicate]

I need to show that the following is an abelian group: $$x*y = \frac{x+y}{xy+1}$$ on the set $\{x \in \Bbb R \,|\, -1 < x < 1\}$. I have been working on this problem, trying to show ...
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1answer
25 views

Showing that compact metrizable abelian group has enough characters

I wanted to know if anybody has some clue as why a compact metrizable abelian group has enough characters to sepatare points. (a character on a topological group is a continuous homomorphism from the ...
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1answer
38 views

Prove that $C_8\times C_2$ has an isomorphic subgroup U and $G/U$ is isomorphic to $C_4$.

Let $G=C_{p^{k_1}}\times C_{p^{k_2}}\times ... \times C_{p^{k_n}}$ an abelian $p$ group, while $k_1,...,k_n\in\mathbb{N}$ and $k_1\geq k_2 \geq ...\geq k_n$. A group $U\cong C_{p^{l_1}}\times ...
2
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2answers
207 views

Number of elements in a finite abelian groups

Is the following true? Let $G$ be a finite abelian group with a minimal generating set $S$. By minimal generating set I mean we cannot reduce the cardinality further. Let $S=\{a_1,a_2,\ldots ,a_k\}$ ...