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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Property of locally cyclic groups

I am having difficulty proving that: A group $G$ is locally cyclic if and only if $G$ is isomorphic to a subgroup of $\mathbb{Q}$ or $\mathbb{Q/Z}$. Is there any easy way to prove it? Thanks.
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0answers
11 views

Automorphism group of a locally cyclic group

I am having difficulty proving that: The automorphism group of a locally cyclic group is commutative. Is there any easy way to prove it? Thanks.
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1answer
469 views

Prove that if a group is nilpotent , then its quotient with its Frattini subgroup is abelian

I know that : 1) Nilpotent group is solvable. 2) Subgroup of a solvable group is solvable. 3) Solvable and simple group is abelian. Now I should use these facts to prove it.
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0answers
14 views

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) [closed]

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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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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0answers
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
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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2answers
118 views

Supose that G is a finite abelian group that does not contain a subgroup isomorphic to $Z_p \oplus Z_p$ for any prime $p$. Prove that G is cyclic.

Supose that G is a finite abelian group that does not contain a subgroup isomorphic to $Z_p \oplus Z_p$ for any prime $p$. Prove that G is cyclic. Attempt: If $G$ is a finite abelian group, then let ...
4
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1answer
196 views

Isomorphic finite abelian groups [duplicate]

Let $G$ and $H$ be finite abelian groups. Show that if for any natural number $n$ the groups $G$ and $H$ have the same number of elements of order $n$, then $G$ and $H$ are isomorphic. I know, ...
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1answer
232 views

Question about Finite Abelian Groups [duplicate]

Let $(G, .)$ be a finite abelian group, $G=\{x_1, ..., x_n\}$ and let $x=x_1. \cdots. x_n$. Show that $x^2=e$ Suppose $G$ has no element of order $2$ or that $G$ has more than one element of order ...
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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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3answers
552 views

On Groups of Order 315 with a unique sylow 3-subgroup .

in Dummit and Foote , an exercise asked me to prove that , if $G$ is a group of order $315$ , $G$ has a normal sylow $3$-subgroup then , $G$ is abelian . this is exercise number $27$ , section $5$ , ...
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1answer
80 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 ...
4
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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 ...
3
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0answers
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 ...
2
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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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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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1answer
104 views

Sum of elements of a finite field

Let $F$ be a finite field and $i$ an integer. Calculate the sum of all the elements of $F$, each raised to the $i$th power. My approach so ...
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0answers
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
63 views

$\ker (T \otimes id_{Z})=\ker(T)\otimes Z$

Does $x\otimes y=0 \implies x=0$ or $y=0$? I don't think so, since its equivalent to $B(x,y)=0$ for some bi-linear form. But my teacher said: $\ker (T \otimes id_{Z})=\ker(T)\otimes Z$ where ...
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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
467 views

Euler-Poincaré characteristic and homology

$\DeclareMathOperator{rk}{\text{rk}}$ $\DeclareMathOperator{im}{\text{im}}$ The problem Let $$C = ( C_n \overset{\partial_n}\to C_{n-1} \overset{\partial_{n-1}}\to \dots \overset{\partial_2}\to C_1 ...
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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 $G$?
2
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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
61 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 ...
6
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1answer
318 views

Why do characters on a subgroup extend to the whole group?

As background, I am trying to do exercise 3.10 in Deitmar's "Principles of Harmonic Analysis." I can do most of the problem but I'm stuck on the third part proving surjectivity. Given a locally ...
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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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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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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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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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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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2answers
428 views

If the tensor power $M^{\otimes n} = 0$, is it possible that $M^{\otimes n-1}$ is nonzero?

Let $M$ be a module over a commutative ring $R$. It is possible that $M \otimes M = 0$ if $M$ is nonzero, for example when $R = \mathbb{Z}$ and $M = \mathbb{Q}/ \mathbb{Z}$. What about when higher ...
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3answers
297 views

How to prove that a group with some properties is abelian?

Let $(G,.)$ be a group and $m,n\in\mathbb Z$ such that $\gcd(m,n)=1 $ and $$ \forall a,b \in G:a^mb^m=b^ma^m$$ $$\forall a,b \in G:a^nb^n=b^na^n.$$ Then how prove $G$ is an abelian group ? Thanks ...
2
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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 ...
1
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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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2answers
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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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 ...
2
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1answer
69 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
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
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 ...
3
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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 + ...
3
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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 $m$ ...
2
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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
90 views

Why is $G$ abelian?

If $|G|=pq^2$ with $p,q$ primes and if $p<q$, with $q\not\equiv\pm1\mod p$, why is $G$ abelian ? The $3^{rd}$ Sylow theorem implies that $n_p|q^2$ and $n_p\equiv 1 \mod p$, By hypothesis, ...
3
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0answers
31 views

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
49 views

Rank-complement subgroup existence

Let $G$ be a finitely generated Abelian group. For each subgroup $H$ of $G$, does there exist another subgroup $K$ of $G$ such that $\text{rank}(G)=\text{rank}(H)+\text{rank}(K)$ and ...
1
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1answer
74 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 ...