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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Isomorphism classes and invariant factors of abelian group

Let $G$ be an abelian group with $ord(G)=3374=2\cdot 7\cdot 241$. Calculate all isomorphism classes with the invariant factors $k_1\ ...k_n$ sucht that $k_i$ divides $k_j$ $(i<j)$. Since ...
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If $a^3=1$, is $G$ abelian?

If $G$ is a group that satisfies $a^3=1$ for every $a\in G$, then is $G$ abelian? This is an exercise I found in Jacobson's Basic Algebra. It is analogous to the question: If $G$ is a group that ...
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1answer
24 views

When this map $f(x)=x^2$ will be a automorphism on $G$, where $G$ is a commutative group.

I know that for a commutative group $G$ the map $f(x)=x^2$ is a homomorphism from $G$ to $G$. My question : When this map $f$ will be a automorphism on $G$, where $G$ is a commutative group. In other ...
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Strange question about free abelian group

I was given the following question for homework, but it makes no sense to me. Since when are free abelian groups constructed w.r.t maps? Isn't the set $S$ all that matters? I don't understand what ...
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121 views

What is the range of this function?

What is the range of $h$? $f(x)=4x+1$ $g(x)=(x-1)/3$ Let $h=\{f^n(g^m(1)):n,m\in\mathbb{N}\geq0\}$ What is the range of $h$? Show that $(2\mathbb{N}-1)\subset H$. ... okay I've done a bit more: ...
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1answer
29 views

Let $G$ be an abelian group can we always construct a quasi projective variety $X$ such that Cl$(X)=G$

Recall that for a quasi projective variety $X$ one can define the Divisor Class Group denoted by Cl$(X)$ Let $G$ be an abelian group. Can we always construct a quasi projective variety $X$ such ...
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34 views

A question about groups

$H$ is called $s$-permutable in $G$ if it permutes with every Sylow subgroup of $G$. $H$ is called $s$-permutably embedded in $G$ if each Sylow subgroup of $H$ is a Sylow subgroup of some ...
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268 views

If G is abelian and simple, then G is cyclic.

True/False: If a group $G$ is abelian and simple,then $G$ is cyclic. Solution: True. If $G$ is an abelian simple group, then $G$ is isomorphic to $\mathbb{Z}_p$ for some prime $p$.
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Q as an additive abelian group has no minimal generating set

$\mathbb Q$ as an additive abelian group has no minimal generating set. I have done this question according to the solution given here. First I took a minimal generating set $S$ of $\mathbb Q$ ...
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2answers
54 views

Show that open interval $(-1,1)$ is isomorphic to $(\mathbb{R},+)$

Define group structure on $G=(-1,1)$ by $$a*b=\frac{a+b}{1+ab}$$ for any $a,b\in G$. Show that $G$ is isomorphic to $\mathbb{R}$ under addition. I've tried the obvious maps $f:G\rightarrow ...
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1answer
19 views

$|\hom(G, \Bbb Z/2)|>1 \iff |G:[G,G]|$ is even

Let $G$ be a finite group. My problem is: $$|\hom(G, \Bbb Z/2)|>1 \iff |G:[G,G]| \mbox{ is even}.$$ I know that $\hom(G, \Bbb Z/2)=\hom(G^{ab}, \Bbb Z/2)$, where $G^{ab}:=G/[G,G]$. If ...
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2answers
45 views

Are these conditions sufficient for a $\mathbb{Z}$-module to be free?

There exist modules over the integers that, like $\mathbb{Q}$, manage to be torsion-free without being free. Ergo, its probably worth looking for conditions $P$ such that "$P$ + torsion-free" is ...
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1answer
23 views

Why if $Z/m\oplus Z/n = Z/mn$ then $(n,m)=1$ [duplicate]

Prove that if $Z/m\oplus Z/n = Z/mn$ then $(n,m)=1$. I have proved the converse, but here there is something I am missing. Hints instead of full answers are appreciated. Thanks.
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1answer
48 views

Determining the number of subgroups of $\Bbb Z_{14} \oplus \Bbb Z_{6}$

I want to determine how many subgroups does the additive group $G:=\Bbb Z_{14} \oplus \Bbb Z_{6}$ have? There are many related posts in our site, for instances: here and there. However, it ...
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38 views

No idea why and cannot prove

Can't find any theorem or helpful ideas that might link to this. In all honestly, I am very lost in this topic. If $H$ is finite abelian group and some $a$ such that $a|\exp(H)$ then $H$ has an ...
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1answer
65 views

$H_1 ,H_2 \unlhd \, G$ with $H_1 \cap H_2 = \{1_G\} $. Prove every two elements in $H_1, H_2$ commute

This is the proof, which I mostly understand except for one bit: You have $h_1 \in H_1$ and $h_2 \in H_2$. We also have $h_1^{-1}(h_2^{-1}h_1h_2) \in H_1$, because $h_2^{-1}h_1h_2 \in ...
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2answers
67 views

All subgroups normal $\implies$ abelian group

This is , I think an easy problem just that I am not getting the catch of it. How to show whether or not the statement is true? All subgroups of a group are normal$\implies$ the group is an abelian ...
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1answer
11 views

Subgroup with order =LCM of two subgroups

The following is a question that was asked by my teacher as a ponder-upon question:to which unfortunately I have not been able to put a single forward step. If an abelian group has subgroups of order ...
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3answers
55 views

Abelian finite group [duplicate]

This is a (maybe be simple) problem from Group Theory, but being a beginner, I am unable to take even a first step forward. Let $G$ be a finite group whose order is not divisible by $3$.Suppose that ...
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1answer
240 views

What is the Betti number of a group?

I'm studying the Fundamental Theorem of finitely generated Abelian group, and it says that the number of factors equal to $\mathbb Z$ (textbook says it is the Betti number of the group) is unique up ...
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3answers
299 views

Check if $(\mathbb Z_7, \odot)$ is an abelian group, issue in finding inverse element

Take $\mathbb Z_7$ and the operation $\odot$ defined on it as follows $\forall a,b \in \mathbb Z_7$: $$\begin{aligned} a \odot b=a+b+3\end{aligned}$$ Check if $(\mathbb Z_7, \odot)$ is a group and ...
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2answers
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Showing that the primary component $G_p$ is a subgroup of $G$

For a finite abelian group $G$ and a prime number $p$ with $p \mid |G|$, we define $G_p$ as the subset of $G$ that contains all elements of $G$ with order $p^k$ for a $k \in \mathbb{N}_0$. We call ...
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1answer
32 views

Prove that the group of the rational points on the conic $u^2-Av^2=1$ is not finitely generated.

This is an exercise from Rational Points on Elliptic Curves by Silverman. Let $H$ be the conic $u^2-Av^2=1$ where $\sqrt{A}\notin \mathbb{Q}$. If $(u_1,v_1), (u_2,v_2)$ are two points in ...
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2answers
46 views

Prove that G is an Abelian Group

Let G = {x in Q : 0≤x<1}. Define the operation on G: a•b = a+b if 0≤a+b<1, a+b-1 if a+b≥1 Prove that (G,*) is an Abelian group. Attempt: (commutativity was easy). For associativity I got a+b≥1 ...
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Groups and queues and stacks

As I review my elementary CS material to prepare for an interview I cannot help but think that I missed a key connection when studying this prior: I think I missed the relationship between operations ...
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0answers
45 views

A Group G is abelian $\Leftrightarrow$ $ Inn(G)$ is a normal subgroup of Sym(G)

First of all I don´t think that this question is answered here If $G$ is non-abelian, then $Inn(G)$ is not a normal subgroup of the group of all bijective mappings $G \to G$ because in my opinion ...
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Criterion for a group $G$ to be abelian using $Inn(G)$ [duplicate]

Let $(G, *)$ be a group. Let $\mathcal S(G)=\{f:G\to G\space|f\space is\space bijective\}$ be the symmetric group and $Inn(G)=\{\kappa_{a}\space|\space a\in G\}$ the inner automorphisms. Now I want to ...
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1answer
40 views

Fundamental group of a tree?

Find the fundamental group of the space $C(T)=\{(x,y) \in T \times T \mid x\neq y\}$. $C(T)=\{(x,y) \in T \times T \mid x\neq y\}$ where $T$ is a graph $T$ is the graph made of $3$ edges with a ...
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15 views

Conjugacy Class Equation for $\mathbb{Z_{25}}$

I'm suposed to find the conjugacy class equation for $\mathbb{Z_{25}}$. Since $\mathbb{Z_{25}}$ is Abelian, that means that $gxg^{-1}=xgg^{-1}=x$ so $Z(\mathbb{Z_{25}})=\mathbb{Z_{25}}$ and every ...
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1answer
33 views

Homomorphism $f:M \to M$ such that $\mathrm{im}(f) = L$ and $f \circ f = f$ where $L \le M$ and $M$ abelian

Let $M$ be an abelian group. For any subgroup $L$ of $M$, can you find a homomorphism $f:M \to M$ such that $\mathrm{im}(f) = L$ and $f \circ f = f$? This was an attempt to find an idempotent ...
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27 views

Does $\phi: A \otimes \mathbb{Q} \to B \otimes \mathbb{Q}$ surj. imply that for $b \in B$, $b = n \phi(a)$ for some $n \in \mathbb{Z}$, $a \in A$?

Let $A$ and $B$ be abelian groups. Suppose that we have a morphism $\phi: A \to B$ and that $\phi \otimes_\mathbb{Z} \mathbb{Q}: A \otimes_\mathbb{Z} \mathbb{Q} \to B \otimes_\mathbb{Z} \mathbb{Q}$ is ...
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1answer
268 views

Prove that $G$ has a element whose order is least common multiple of $m$ and $n$. [duplicate]

Let $G$ be an abelian group and suppose that $G$ has elements of order $m$ and $n$ respectively. Prove that $G$ has an element of order $\mathrm{lcm}[m,n]$
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42 views

Why abelian groups instead of modules in Algebraic Topology

I am studying Algebraic Topology, homology and cohomology to be concrete. I am reading\working through Hatcher, Rotman, Harper and sometimes I combine them with other books when none of them give a ...
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1answer
57 views

Non-abelian group which squares to equal the identity element?

Does there exist a non-abelian group $G=\{e,g_1,g_2,...,g_n\}$ with order $n+1$ s.t. \begin{align} (g_1 \dots g_n)^2 = e \end{align} Also, does this change if we say that every element in $G$ is its ...
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1answer
31 views

Regarding those $\mathbb{Z}$-modules whose every finite subset generates a finite submodule.

Let $X$ denote a $\mathbb{Z}$-module (aka an abelian group). Then $X$ may or may not satisfy: $(*)$ for all finite sets $F \subseteq X$, the module generated by $F$ is finite. This properly ...
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2answers
60 views

Can Lagrange's Theorem for algebraic structure apply here?

For a positive integer $n$ let $Φ(n)$ denote the number of elements $r∈\mathbb Z_n$ such that $\gcd(r,n)=1$. Show $Φ(mn)=Φ(m)Φ(n)$ for all $m, n∈\mathbb N$ such that $\gcd(m,n)=1$. The only thing I ...
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1answer
86 views

Direct proof that infinite product of copies of $\mathbb{Z}$ is not projective

It is well-known that the abelian group $$A = \prod_{n=1}^\infty \mathbb{Z}$$ is not free (see, for example this MO question), and that over a PID being free is equivalent to being projective (see ...
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1answer
30 views

Proving that a normal, abelian subgroup of G is in the center of G if |G/N| and |Aut(N)| are relatively prime.

I was trying to prove that a normal, abelian subgroup of $G$, $N$ is in the center of $G$ given that $|\operatorname{Aut}(N)|$ and $|G/N|$ are relatively prime. The official question: Let $N$ be ...
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2answers
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Does $(\mathbb{Z} \times \mathbb{Q})/M$ have any element of infinite order? [on hold]

Let $\mathbb{Z} \times \mathbb{Q}$ be the group of ordered pairs $(x, y)$ with $x \in \mathbb{Z}, y \in \mathbb{Q}$ under component-wise addition. Fix $m \in \mathbb{Q}$ and let $M \subset \mathbb{Z} ...
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When is the automorphism group $\text{Aut }G$ cyclic?

Let $G$ be a finite group. Under which conditions on $G$ is the automorphism group $\text{Aut }G$ cyclic? More precisely, does "$G$ is abelian" or "$G$ is cyclic" imply "$\text{Aut }G$ is cyclic"?
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1answer
70 views

$G$ be an infinite group such that $|Aut (G)|=2$ , then is $G$ cyclic ?

Let $G$ be an infinite group such that $|Aut (G)|=2$ , then is $G$ cyclic ? Since $Aut(G)$ is cyclic here , I know that $G$ is abelian , but this is as far as I can get . Please help . Thanks in ...
2
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1answer
38 views

$G$ is an infinite abelian group such that $G \cong H$ for every non trivial subgroup $H$ of $G$ , then is $G$ cyclic?

If $G$ is an infinite abelian group such that $G \cong H$ for every non trivial subgroup $H$ of $G$ , then is $G$ cyclic , or equivalently asking , then is $[G:H]$ finite for every non trivial ...
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1answer
211 views

A group whose automorphism group is cyclic

Is there an Abelian group $A$ which is not locally cyclic whose automorphism group is cyclic ?
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1answer
191 views

On groups with none of their quotient groups divisible [closed]

Does there exist a group $G$ that satisfies the following conditions: Any proper subgroup of $G$ is contained in a maximal subgroup. There is some $N\unlhd G$ such that $\frac{G}{N}$ is divisible. ...
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1answer
38 views

Identify the abelian group that has the given presentation matrix

For the presentation matrices $$ \begin{bmatrix} 0 \\ 5\\ \end{bmatrix} , \begin{bmatrix} 1 & 0 \\ 0 & 1\\ 0 & 0 \end{bmatrix}$$ identify the abelian group they represent. For the ...
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On describing a sort of “well-behaved” subgroups of a free abelian group.

I found this question when I tried to figure out what kind of subgroups of a free abelian group behave just as well as in the finite generated case. Let $M$ be an free abelian group, $N$ a subgroup ...
8
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1answer
75 views

Automorphisms of infinite abelian groups

It is well-known that the map $Aut$ from the class of groups to itself has fixed points. For $n \neq 2$ or $6$, $Aut(S_n) \cong S_n$, $Aut(D_4) \cong D_4$ and if $G$ is a finite non-abelian simple ...
4
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1answer
62 views

Why do the characters of an abelian group form a group?

I was reading through Serre's Linear Representation Theory book and encountered a question to show that the set of all irreducible characters of an abelian group form a group. The proof of closure ...
2
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2answers
24 views

Question to the proof of: Let $A$ be a finite abelian group and let $g \in A$. Suppose that $\chi(g)=1$ for every $\chi \in \hat A$. Then $g=1$.

Good day, Currently I am working with the book "A First Course in Harmonic Analysis" by A. Deitmar and I am stuck in the beginning of Chapter 5 on the proof to Lemma 5.1.5. I am repeating the few ...