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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The generator of a cyclic quotient group

(I'll use (a) to denote the subgroup of G generated by a) Let G be a finite abelian group of order n. Let a be an element of G of order k. We can easily see that (a) is a normal subgroup of G. If we ...
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2answers
28 views

Prove $G\cong H\oplus \Bbb{Z}^{k}$.

Let $G$ be an abelian group and let $H$ be a subgroup. Let $G/H\cong \Bbb{Z}^{k}$. Prove $G\cong H\oplus \Bbb{Z}^{k}$. What I did so far is: there is an epimorphism from $G$ to $\Bbb{Z}^{k}$ such ...
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1answer
29 views

Questions in Abstract Algebra

I have two question which I couldn't solve: Let $G$ be a group of size $40$. a. Show the $5$-Sylow subgroup in $G$ is Normal - this part was easy, I just showed that $n5=1$ and then $P5$ is ...
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1answer
30 views

Universality of tensor product from monoidal structure

As a follow-up to this previous question of mine, I'm trying to understand how to obtain tensor products from internal homs. I'm having a lot of difficulties and have found myself stuck already in ...
4
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0answers
34 views

Automorphisms of Abelian groups

Let $A$ be a free Abelian group and $N$ a characteristic subgroup of $A$ such that $A/N$ is finite. I also know that $Aut(A/N)$ and $Aut(N)$ are both finite. I have to prove that $Aut(A)$ is finite. ...
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24 views

The abelianness of the quotient group of an abelian group.

I am working on an assignment for my abstract algebra class. The question states: Let $A$ be an abelian group and let $B$ be a subgroup of $A$. Prove that $A/B$ is abelian. I was under the ...
2
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2answers
33 views

Finding an order of a coset in $A/B$ where $A$ is a free abelian group and $B$ is a subgroup.

Let $A$ be a free abelian group with basis $x_1,x_2,x_3$ and let $B$ be a subgroup of A generated by $x_1+x_2+4x_3, 2x_1-x_1+2x_3$. In the group $A/B$ find the order of the coset $(x_1+2x_3)+B$. How ...
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2answers
82 views

Order of any element divides the largest order.

Let $A$ be a finite Abelian group and let $k$ be the largest order of elements in A. Prove that the order of every element divides $k$. This is my attempt, I sense there is something wrong\incorrect ...
2
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2answers
52 views

Subgroup of $\Bbb {Z}_m \oplus \Bbb {Z}_n$ where $(m,n)=1$.

Let $m,n>1$, $(m,n)=1$. Prove that every subgroup $H$ of $\Bbb {Z}_m \oplus \Bbb {Z}_n$ is $H=A\oplus B$ where $A=H\cap \Bbb {Z}_n$ and $B=H\cap \Bbb {Z}_m$. First attempt: $G=\Bbb {Z}_m \oplus ...
3
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2answers
82 views

Does there exist an abelian group with insoluable word problem?

Does there exist an abelian group with recursively enumerable presentation and insoluble word problem? My gut says "of course not!". However, my mind keeps saying "but...doesn't $\mathbb{R}$ have ...
4
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2answers
30 views

Existence of non-split sequence

Let $G$ be an abelian group such that $G$ contains non-zero elements of finite order. Why there exists some short exact non-split sequence: $0 \rightarrow \mathbb{Z} \rightarrow H \rightarrow G ...
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1answer
24 views

Characterizing the Prüfer $p$-group

I've been trying to solve these questions for the past few hours with no luck: If $G$ is an infinite abelian group all of whose proper subgroups are finite, then $G$ is a Prüfer $p$-group for some ...
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0answers
12 views

if A/B is a torsion group then rank(A)=rank(B)

Let $A$ a finitely generated abelian group, $B\subset A$ a subgroup such that $A/B$ is a torsion group. Then $rank(A)=rank(B)$
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1answer
54 views

Is the group $G =\{a+b\sqrt{2}|a,b \in \mathbb{Z}\}$ cyclic?

$G = \{a+b\sqrt{2}|a,b \in \mathbb{Z}\}$ under addition: I am going to say it's not cyclic because a,b can be distinct. I tried finding a generator.
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2answers
41 views

Law of Exponents for Abelian Groups

Let $a$ and $b$ be elements of an Abelian group and let $n$ be any positive integer. Show that $(ab)^n = a^nb^n$. Is this also true for non-Abelian groups?
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Example where a finite group $G$ of order $n$ has no subgroup of order $m$

Using the Fundamental Theorem of Abelian Groups, one can prove that if $G$ is a finite abelian group of order $n$ such that $m$ is a positive integer that divides $n$, then $G$ contains a subgroup of ...
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1answer
68 views

Group theory- rank of a group. What am I doing wrong?

I was given a question: Let $n\in \mathbb{N}$ and let $A$ and $B$ groups, both isomorphic to $\mathbb{Z}^n$. Let $f:A \to B$ be a surjective homomorphism. Prove $f$ is an isomorphism. Here's my ...
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0answers
39 views

Graphing elliptical curves based on group operation

I just found this and it blew my mind (he gives an elliptical curve to do multiplication). If I understand correctly (from reading the link and other things) the Abelian group he is using is ...
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0answers
39 views

Proving that product of two quotients = a certain quotient group

Question I am producing certain given conditions from a paper and a certain fact(stated in the paper) that I need to prove using those conditions. I am converting this problem into a general group ...
4
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1answer
47 views

If $A\oplus B\cong A\oplus C$ then $B\cong C$

Let $A,B,C$ be finitely generated abelian groups, and $A\oplus B\cong A\oplus C$. Prove that $B\cong C$. I know that it follows from the fundamental theorem of finite abelian groups, but I have ...
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2answers
43 views

To make $(K_4,+)$ ( the Klein-4 group ) a ring

How can we define an operation $.$ such that the Klein-4 group $(K_4,+)$ becomes a ring $(K_4,+,.)$ ?
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1answer
79 views

Bijection between different sets of homomorphisms.

Let $G$ be a group and let $A$ be and abelian group. Let $G'$ be the commutator of $G$. Prove there is a bijection between the set of homomorphisms $f:G\rightarrow A$ to the set of homomorphisms ...
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1answer
27 views

Example of a non-abelain finite group $G$ with $G/N$ abelian and infinite group $G$ with $G/N$ finite

Have not been able to think of a examples with the following properties: Example of a non-abelian finite group $G$ with property that $G/N$ is abelian for every non-trivial normal subgroup $N$ of ...
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1answer
41 views

Question about corollary 2.1.6 in Cohen's Number Theory vol. 1.

Corollary 2.1.6. Let $ V \in \mathbb{Z}^n $ be a column vector of $ n $ globally coprime integers. There exists an integral matrix $ A \in GL_n (\mathbb {Z}) $ ( in other words with determinant $1$ ) ...
2
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2answers
54 views

A rank-nullity theorem between $\mathbb Z^n$ and $\mathbb Z^k$ [duplicate]

I think this is correct: If $\phi:\mathbb Z^{n}\to\mathbb Z^{k}$ is a group homomorphism then $n=\operatorname{rank}\operatorname{im}\phi+\operatorname{rank}\ker\phi$. Here is my attempt at a ...
2
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1answer
44 views

Prove that $A \approx_n B$ if and only if $A \cap B$ is of finite index both in $A$ and in $B$

Suppose that $A,B$ are torsion-free abelian groups of rank $n>1$. Then $A$ is said to be quasi-contained in $B$, written $A \mathrel{ \prec_n} B$, if there exists an integer $m>0$ such that ...
0
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1answer
23 views

Text for study of subgroup lattices of finite abelian groups.

I want to study the subgroup lattice of a finite abelian group. I have found a text on the subject: Subgroup Lattices of Groups by Roland Schmidt, de Gruyter 1994. This book is about subgroups of any ...
1
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1answer
60 views

Automorphisms on groups

Can someone clarificate how is defined $\text{Aut}(\Bbb Z_n,+)$ and how we can find it? I understand that $\text{Aut}(\Bbb Z)$ are the functions $x\to x$ and $x\to -x$ because a generator image must ...
0
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2answers
38 views

Finite index subgroup $G$ of $\mathbb{Z}_p$ is open.

Suppose $[\mathbb{Z}_p:G] = n <\infty$. Write $n = p^km$ with $p\nmid m$. The idea is to show that $p^k\mathbb{Z}_p = n\mathbb{Z}_p \subseteq G$, after which I am done, since for any $x\in G$ we ...
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0answers
21 views

A Question About the Maximal Subset of a set of Generators of a Finitely Generated Torsion Free Abelian Group Forming a Basis

First, I should apologize for the title, I don't know how to put it more succinctly. Its more grandiose incarnation would be: 'Why Does the Maximal Subset of a set of Generators of a Finitely ...
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2answers
111 views

How can I understand $\Bbb Z\times \Bbb Z/2\Bbb Z$

This may be stupid request, but I would like to have a intuition for the group $\Bbb Z\times \Bbb Z/2\Bbb Z$ in terms of 'real' objects. 'Real' could mean geometric but not necessarily. I perhaps what ...
4
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2answers
92 views

Let $G$ be a group of order 315 with a normal 3-Sylow subgroup. Prove $G$ is abelian.

I know this it a prevalent question, I really do. It's just that every proof requires using Automorphisms groups about which we were barely taught. I can't start learning everything about ...
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1answer
49 views

What does $d_{n+1}\circ{d_{n}}=0$ mean in the definition of a chain complex?

According to the Wiki article on chain complexes, a chain complex $(A_{\bullet},d_{\bullet})$ is a sequence of abelian groups or modules connected by homomorphisms such that $d_{n+1}\circ{d_{n}}=0$. ...
2
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1answer
73 views

Let $G$ to be abelian group and $|G|=mn$ when $(m,n)=1$. $G_m=\{g\mid g^m=e\}$,$G_n=\{g\mid g^n=e\}$, prove isomorphism

I want to prove $ f:G_n\times G_m\rightarrow G$ when $f(g,h)=gh $ is an isomorphism First of all I showed that $G_m,G_n$ are subgroups of $G$ (easy) Now I want to show that for every $ a,b, \in ...
3
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4answers
68 views

$O(G)=p^2 $ ,and p is prime, it is also known that $|Z(G)|>1 $. proof that G is abelian

We know that $Z(G)<G,\;$ then $O(Z(G)) \mid O(G). $ If $\;O(Z(G))= p^2, $ then $\;Z(G)=G$ and we are done. Now, if $O(Z(G))= p,\,$ how can I prove that $G$ is abelian ? Is it by proving that ...
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0answers
44 views

Prove that if $A,B$ are finite Abelian groups and for every $n$ they have the same amount of elements of order $n$, then $A \simeq B$ [duplicate]

Prove that if $A,B$ are finite Abelian groups and for every $n$ they have the same amount of elements of order $n$, then $A \simeq B$. I know I have to use primary decomposition, but am not sure how ...
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3answers
43 views

If $A$ is an Abelian group and $B < A$, $ A \simeq B \simeq \mathbb Z^n$ for some natural $n$. Prove that $mA \subset B $ for some $m$.

If $A$ is an Abelian group and $B < A$, $ A \simeq B \simeq \mathbb Z^n$ for some natural $n$. Prove that $mA \subset B $ for some $m$. I know this has something to do with the fact that there ...
1
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1answer
45 views

Finitely generated abelian groups isomorphism

Got this on a home assignment and I don't have a clue... How do I determine if $\mathbb{Z}_{12}\times\mathbb{Z}_{18}$ and $\mathbb{Z}_{6}\times\mathbb{Z}_{36}$ are isomorphic? Any hints will be very ...
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1answer
48 views

How to list all subgroup info of a (n abelian) group with Sage?

Using the Magma calculator at http://magma.maths.usyd.edu.au/calc/ I listed all subgroups of C3XC3 : ...
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2answers
70 views

If $g^2 = e$ for all $g \in G$, then $G$ is abelian [duplicate]

Let $G$ be a group. Prove that $g^2 = e$ for all $g \in G$, then $G$ is abelian. ($e$ is the identity element.) My Solution: Let $a,b \in G$. Then $a(ab)b = a^2b^2 = e^2 =e$. Now I tried to reverse ...
4
votes
2answers
104 views

Finite Abelian groups with the same number of elements for all orders are isomorphic [closed]

Let $A$ and $B$ be finite abelian groups. Suppose that for every natural number $m$, the number of elements of order $m$ in $A$ is equal to the number of elements of order $m$ in $B$. Prove that $A$ ...
3
votes
1answer
169 views

Free abelian group, matrix representation.

Let $n$ be a positive integer and let $A$ be a free abelian group and let $\{e_1,\dots,e_n\}$ be a basis of $A$. Let $B$ be a subgroup generated by $\{v_1,\dots,v_n\}$ and $M=(m_{ij})$ be a matrix ...
0
votes
1answer
35 views

Subgroups isomorphic to $\mathbb{Z}_{5}\oplus\mathbb{Z}_{5} $

Let $A=\mathbb{Z}_{360}\oplus\mathbb{Z}_{150}\oplus\mathbb{Z}_{75}\oplus\mathbb{Z}_{3}$. I need to calculate the number of subgroups of $A$ which is isomorphic to $\mathbb{Z}_{5}\oplus\mathbb{Z}_{5}$ ...
4
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1answer
88 views

A question about a free abelian finitely generated group.

I am having a hard time solving this and it is really confusing. I don't have enough schema, which makes it problematic. Let $A$ be a finitely generated free abelian group and $B$ is a subgroup of ...
4
votes
1answer
58 views

Show that there is a positive integer $n$ such that $G \cong \ker(\phi^{n}) \times \phi^{n}(G)$

Let $G$ be a finite abelian group and let $\phi: G \rightarrow G$ be a group homomorphism. I am trying to show that there is a positive integer $n$ such that $G \cong \ker(\phi^{n}) \times ...
0
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1answer
75 views

I don't understand this notation- abelian groups

May be a stupid question but is $(\mathbb{Z}^n)_p \equiv \mathbb{Z}^n/(\mathbb{Z}p)^n$ (when $p$ is a prime)??
4
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0answers
81 views

A question about the automorphism group of $\mathbb{Z}_{2} \times \mathbb{Z}_{4}$

I wanted to clarify some confusion I was having on the automorphism group of $\mathbb{Z}_{2} \times \mathbb{Z}_{4}$, which I call $Aut(\mathbb{Z}_{2} \times \mathbb{Z}_{4})$. I considered the ...
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3answers
128 views

Two problems in Group theorem related to Sylow's theorem(maybe)

Prove that any subgroup of order $ p^{n-1} $ in a group $G$ of order $p^{n}$, p a prime number, is normal in $G$. $(a)$ Prove that a group of order 28 has a normal subgroup of order 7. To deal ...
3
votes
2answers
49 views

$\Bbb{Q}$ is not a finitely generated $\Bbb{Z}$-module

I'm trying to show that $\Bbb{Q}$ is not a finitely generated $\Bbb{Z}$-module. Assume to the contrary that $$\Bbb{Q}=\Bbb{Z}\dfrac{a_1}{b_1}+...+\Bbb{Z}\dfrac{a_n}{b_n}$$ where $a_i,b_i\in\Bbb{Z}$. ...
3
votes
4answers
170 views

How to show $\mathbf{Q} $ is not free

We know that torsion free plus finitely generated $\rightarrow$ free and that $\mathbf{Q}$ is torsion free is easy. But how to show $\mathbf{Q}$ is not finitely generated and not free?