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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Center of an abelian group

Prove if $G$ is non abelian group, then exists an abelian subgroup $H$ which contains $Z(G)$ and $H≠Z(G)$.
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1answer
22 views

Inverse element of the abelian group (P(M),symmetric difference)

What would be the inverse element in this abelian group: $(P(M),\triangle)$? I know the neutral element is the empty set and I thought the inverse element would be $A^{c}$ for every $A$. Turns out ...
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1answer
34 views

Why homomorph and not isomorph?

Why are the groups $\mathbb{R},+$ and $\mathbb{R}_0^+,*$ homomorph, their mapping function being $ f: x \rightarrow e^x $? Why is this not an isomorphism?
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1answer
45 views

finetely generated Abelien -by-nilpotent group

Let G finetely generated Abelien -by-nilpotent group (i.e there existe a abelien subgroup H in G and G/H is nilpotent )With each of its two-generator is nilpotent-by-finite show that G is ...
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1answer
67 views

Show there is no non-abelian group of order 9 [duplicate]

I want to show there is no non-abelian group of order 9. How should I attempt this?
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1answer
117 views

For abelian groups: does knowing $\text{Hom}(X,Z)$ for all $Z$ suffice to determine $X$?

Let $X$ and $Y$ be abelian groups. Suppose $\text{Hom}(X,Z)\cong \text{Hom}(Y,Z)$ for all abelian groups $Z$. Does it follow that $X \cong Y$? It has been answered before that this is true if the ...
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2answers
59 views

Non abelian subgroup of a abelian group.

What is the relationship between abelian subgroup of a non-abelian group(when exist, example, theorem)?? any thing such link regarding the question would help. ...
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2answers
28 views

Order and Least Common Multiple Abelian Question

\item Let $G$ be an abelian group and let $x, y\in G$ be elements so that $o(x)=m$ and $o(y)=n$. Show that $o(xy)=\frac{mn}{(m,n)}$. (Note that this is the least common multiple of $m$ and $n$) Is ...
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1answer
63 views

How is the entire $SO(2)$ group the standard rotation matrix?

In a book I am using, the following is presented, $$\mathcal{R}(\phi) = \begin{pmatrix} \cos (\phi ) &\sin (\phi ) \\ -\sin (\phi ) &\cos (\phi )\end{pmatrix}$$ The group's name is ...
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1answer
27 views

Subgroups of group of characters of a finite abelian group

Let G be a finite abelian group, H a subgroup of the group of characters of G. Is it true that H is the group of characters of some quotient group of G? Thanks for any help.
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1answer
27 views

$p^nm$ group, element of order $m$

Let $p$ be an abelian group of order $p^nm$, $p$ prime, and $p$ does not divide $m$. Is it true that the group must contain an element of order $m$, or a multiple of $m$? If yes, how to prove it? If ...
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5answers
70 views

If $a, b$ are in group and $ab$ has finite order $n$, why does $ba$ have order $n$ as well? [duplicate]

If $a, b$ are in group and $ab$ has finite order $n$, why does $ba$ have order $n$ as well? Since $(ab)^n=e$, I get $(b)(ab)^n(a)= ba$. This means that $(ba)^{n+1}=ba$, and $(ba)^n=e$. But, I ...
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0answers
47 views

Let $G$ be a group and let $\Phi\colon G \to G$ be an isomorphism. Define $H = \{ a \in G\ |\ \Phi(a) = a^{-1} \}$ [duplicate]

It asks to prove that if $H$ is a subgroup of $G$, then $G$ is abelian. Solution: I showed that for every $a$ and $b$ in $H$, $a$ and $b$ commute. But how do I generalize to elements in $G$ NOT in ...
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2answers
53 views

Prove rk$B$ $\le$ rk$A$ where A and B are free, abelian and finitely generated groups.

Let $A$ and $B$ be free abelian, finitely generated groups. Let $f:A \to B$ be an epimorphism. Prove rk$B$ $\le$ rk$A$. I could really use a verification. That is a question from my exam today. ...
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0answers
59 views

$gcd(|G|, |Aut(G)|)=1$ means G is abelian?

Prove the following assuming that G is finite group with $gcd(|G|, |Aut(G)|)=1$ a)G is abelian (done) b) Every Sylow subgroup of G is cyclic of prime order. G is abelian than every sylow unique, ...
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1answer
46 views

Prove that $G$ has a subgroup $L$ and $|L|=mn$

For $G$ is an abelian group, $H,K$ are subgroups of $G$ and $|H|=n,|K|=m$. Prove that $G$ has a subgroup $L$ and $|L|=mn$ In cases $H\cap K=\{e\}$ use Lagrange theorem we can show that $|HK|=mn$ but ...
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76 views

Abelian groups whose automorphism group is a $p$ group

$\def\Aut{\operatorname{Aut}}$ Let $G$ be a finite abelian group such that $\Aut(G)$ is an $p$ group ,that is, $|\Aut(G)|=p^n$ . Then can we determine the cyclic decomposition of $G$ or at least the ...
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2answers
36 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
35 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
35 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 ...
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0answers
36 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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2answers
26 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 ...
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2answers
41 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
91 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 ...
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2answers
56 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 ...
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2answers
101 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 ...
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2answers
38 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
83 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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16 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
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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
91 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
87 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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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
43 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
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
51 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
94 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
43 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 $1$ ) ...
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2answers
67 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 ...
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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
65 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
54 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
38 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
120 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 ...
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2answers
115 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
53 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 $d_{n+1}\circ{d_{n}}=0$. ...
4
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1answer
84 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, ...
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 ...