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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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
80 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 ...
5
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0answers
49 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 ...
0
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0answers
41 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
votes
1answer
49 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
50 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 $(K_4,+,.)$ ?
1
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1answer
90 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
39 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
44 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
66 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 ...
0
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1answer
29 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 ...
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1answer
64 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 ...
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2answers
52 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
37 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 ...
0
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2answers
118 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
109 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
50 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 $d_{n+1}\circ{d_{n}}=0$. ...
3
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1answer
80 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
votes
4answers
70 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
44 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 ...
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1answer
48 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
78 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
87 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
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2answers
125 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
185 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
39 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
92 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
66 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
79 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)??
5
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106 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
146 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 ...
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2answers
75 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
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4answers
176 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?
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2answers
94 views

Number of subgroup of order $p^2$ in $\mathbb{Z}_{p^{2}} \times \mathbb{Z}_{p^{2}}$

Let $G = \mathbb{Z}_{p^{2}} \times \mathbb{Z}_{p^{2}}$. How many subgroups does $G$ has of order $p^2$? I know there are only 2 cases of the subgroup H, H can be isomorphic to $Z_{p^{2}}$ or $ ...
0
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0answers
13 views

Reference for work on abelian divisible groups $G$ such that for every $n \in \mathbb N , g \in G , \exists$ unique $x \in G$ such that $g=x^n$

Is there any work or reference in the literature about those abelian divisible groups $G$ such that for every $n \in \mathbb N , g \in G , \exists$ unique $x \in G$ such that $g=x^n$ ; I think then I ...
0
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1answer
36 views

proving a group

Consider the group ($[0; 2\pi)$;$\oplus_{2\pi}$) where $\oplus_{2\pi}$ means addition modulo $2\pi$. Define $S_1$ as the following subset of $\mathbb{C}$ $S_1$ = {$z \in \mathbb{C}; z = e^{i\theta}; ...
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2answers
81 views

Groups such that inclusion on collection of all its subgroup is a total order

Characterize the Groups with the following property: Suppose G is any group such that for any two subgroups, H and K either H $\subseteq$K or K $\subseteq$ H. Now what can we tell about cardinality, ...
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0answers
47 views

Show for $x\in G$ written as product of $hk$.

Show for $x\in G$ written as product of $hk$ for $h \in H$ and $k\in K$. Let $G$ be of order $p^km$ for $p$ is prime and does not divide $m$. $H=(x\in G\mid x^{p^k}=e)$ and $K=(x\mid x^m=e)$. G is not ...
2
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1answer
60 views

For $G$ an abelian group and $H$ a subgroup, is $[G : H]$ the smallest positive integer $n$ such that $ng \in H$ for all $g \in G$?

Let $G$ be an abelian group and $H$ a subgroup. What is the smallest positive integer $n$ such that $ng \in H$ for all $g \in G$? Is it $[G : H]$, or can it be strictly smaller (a divisor of $[G : ...
1
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1answer
58 views

Odd order n smaller than 27

I have a group $G$ that is a group of matrices of the form $$\left( \begin{array}{ccc} 1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \end{array} \right)$$ where $a,b,c \in \Bbb Z_3$. ...
1
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2answers
73 views

Understanding the definition of tensor product as a quotient of a free abelian group

I've been give the Definition: Let F be a free abelian group with a basis $X$ such that. $$F = \langle A\times B\mid \emptyset \rangle $$ Let $f$ be a subgroup of $F$ generated by the ...
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3answers
182 views

Prove a quotient group is abelian

Let $G$ be a group with a normal subgroup $M$ such that $G/M$ is abelian. Let $N\geq M$ and $N \unlhd G$. Show $G/N$ is abelian. My attempt: To show that $G/N$ is abelian, we need to show that for ...
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8answers
474 views

Every infinite abelian group has at least one element of infinite order?

Is the statement true? Every infinite abelian group has at least one element of infinite order. I am searching for an infinite abelian group with all elements having finite order. Please help me ...
3
votes
1answer
131 views

Which of the following abelian groups are cyclic groups?

Given the abelian groups of order $7425$: $$Z_{33} \times Z_{15} \times Z_{15} , \ Z_{25} \times Z_{297} , \ Z_{45} \times Z_{165} , Z_{55}\times Z_9 \times Z_{15}$$ Which of these groups, if ...
0
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2answers
45 views

let $G$ to be finite abelian group of order O(G), let n to be prime number and (O(G),n)=1 prove that $g=x^n$ for any $ x \in G$

let $G$ to be finite abelian group of order O(G), let n to be prime number and (O(G),n)=1 prove that $\forall g \in G$ we can write $g=x^n$ for any $ x \in G$
3
votes
4answers
48 views

$G$ is a finite abelian group and $m:=\max \{o(x):x \in G\}$ , then is it true that $o(x)|m , \forall x \in G$?

If $G$ is a finite abelian group and $m:=\max \{o(x):x \in G\}$ , then is it true that $o(x)|m , \forall x \in G$ ?
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0answers
49 views

If $X$ is an Abelian group, then $\ker_X : \mathrm{Cong}(X) \rightarrow \mathrm{Sub}(X)$ is a bijection. Is there a partial converse?

(All monoids are written additively in this question, even the non-commutative ones.) Given a monoid $X$, write $\mathrm{Sub}(X)$ for the lattice of submonoids of $X$, and write $\mathrm{Cong}(X)$ ...
2
votes
2answers
59 views

A group in between the commutator subgroup and the original group must be normal

Let $C$ be the commutator subgroup of a group $G$. then by some easy arguments, we know that $1$. $C$ is normal in $G$ $2$. $G/C$ is abelian $3$. If $N$ is normal in $G$ and $G/N$ is abelian, then ...
5
votes
1answer
100 views

Is there a homomorphism from a full product of finite cyclic groups onto $\mathbb Z$?

Trying to answer this question, I encountered the following question, the answer to which should be known but it is hard to Google, so I did not find it. Let $G=\prod_{n\in\mathbb N}\mathbb Z_n$ be ...