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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58 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)??
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0answers
66 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
114 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
44 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
160 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
79 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 $ ...
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0answers
9 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
34 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}; ...
0
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1answer
33 views

commuting subsets in a group

For a countable infinite discrete group $G$, consider the following three properties. (P1) $G$ is abelian. (P2) For any finite subset $K$ of $G$, there exists an element $s\in G$, such that the two ...
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2answers
57 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
39 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
54 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
54 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 $a,b,c \in \Bbb Z_3$. ...
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2answers
35 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 ...
3
votes
3answers
142 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
364 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
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1answer
101 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
38 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$
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4answers
45 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
41 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
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2answers
46 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 ...
4
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1answer
84 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 ...
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2answers
49 views

Prove every subgroup S of a finitely generated abelian group G is itself finitely generated.

Call a group G finitely generated if there is a finitely subset X$\subseteq$G with G=$<X>$. Prove that every subgroup S of a finitely generated abelian group G is itself finitely generated. I ...
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0answers
33 views

Order of center of a p-group deduce abelian [duplicate]

Let $G$ be a group of order $p^n$, $p$ a prime. Suppose the center of G has order at least $p^{(n−1)}$. Prove that G is abelian. Attempt: use the class equation $|G|=|Z(G)|+ \sum_{i \in ...
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3answers
70 views

$G/Z(G)$ is cyclic useful for proving groups abelian?

It's a common exercise to prove in an abstract algebra book that if $G/Z(G)$ is cyclic then $G$ must be abelian. But I've always found the exercise strange because if $G$ is abelian then $Z(G)=G$ and ...
2
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0answers
20 views

Direct product of groups and isomorphism [duplicate]

Let $A, B, C$ three groups such that $A \times C \cong B \times C$. I already know that if $A, B$ and $C$ are abelian and finite, then $A \cong B$. I think this result does not hold anymore if they ...
0
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1answer
30 views

Finding/Creating a Modern Algebra theorem

The question I'm trying to prove is this one: The subgroup $<G,S>$ generated by $G$ and $S$ is abelian and of order $9$. My Work: $G=(123)(456)(789)\ \text{and} \ S=(147)(258)(369)$ ...
0
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0answers
45 views

group without involution is 2-divisible

Let $G$ be an arbitrary torsion group without involutions. Show that $G$ is 2-divisible. I think it is enough to show $G$=$2G$ but i can't show why $2G$ can't be proper subgroups of $G$ ? Please ...
8
votes
1answer
94 views

Maps to all finite cyclic groups factor implies map to integers factors

Let $G$, $H$ be groups (we lose nothing here if we assume they're abelian), let $f:G\to H$ and $g:G\to \mathbb{Z}$ be homomorphisms. This last map gives us homomorphisms $g_n:G\to {\mathbb{Z}}/{n ...
7
votes
1answer
176 views

Five lemma: unique isomorphism?

Consider the Five lemma with abelian groups. If $l$, $m$, $p$, and $q$ are isomorphisms, then $n$ is an isomorphism. Let $n'\colon C\to C'$ be a second homomorphism such that $ n' \circ g=s\circ m$ ...
1
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1answer
52 views

A4 is a normal subgroup of A5

The problem is that: How to check, if $A_{4}$ is normal (or not) subgroup of $A_{5}$? We know that $|A_{5}|=60$ - i suppose that we shouldn't find all left and right conjugate classes, because it's a ...
0
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1answer
24 views

Existence of integer $n > 2$ such that for any abelian group $G$ , $G_n:=\{e\} \cup \{a \in G :o(a)=n \} $ is a subgroup of $G$ [closed]

Does there exist an integer $n > 2$ such that for any abelian group ( or at-least any finite abelian group ) $G$ , the set $G_n:=\{e\} \cup \{a \in G :o(a)=n \} $ is a subgroup of $G$ ?
1
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1answer
29 views

Automorphisms of the Prufer group

Let $p$ be a prime number. Can you give me a few examples of automorphisms of $\Bbb Z_{p^\infty}$ other than the identity function? I'm looking for an elemetary way to construct them. It can be ...
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2answers
39 views

Determing whether (S, *) is an Abelian group

Given a set defined as $S=\{(a,b) | a,b \in \mathbb{Q} \land a^2+b^2=1 \}$ and a binary operation $*$ defined as $(\forall(a,b),(c,d) \in S) ((a,b)*(c,d) = (ac-bd, bc+ad))$, determine whether $(S,*)$ ...
0
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1answer
19 views

for abelian $A\ncong \mathbb Z_2 ,\{e\} $ to finde a automorphism that is not trivial

let $A\ncong \mathbb Z_2 , \{e\}$ abelian group, i want to find a automorphism $\varphi\neq Id_A$. i tried to define it such that for every $a\in A $ , $\varphi (a)=-a$. this definition will do ...
8
votes
0answers
172 views

Writing this $G$-set explicitly as union of orbits

Let $G$ be a finite abelian group, and let $A$ and $B$ be subgroups. I'm interested in $G/A\times G/B$ with its natural $G$-set structure. In $G/A\times G/B$, the stabilizer of any element is $A\cap ...
2
votes
1answer
64 views

Every nonabelian group of order divisible by 6 contains a subgroup of order 6

I have a question I was hoping for help on: Prove or disprove every nonabelian group of order divisible by 6 contains a subgroup of order 6 I would guess that this statement is true based on a ...
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2answers
46 views

Showing that the product group of $G$ and $H$ satisfies the universal property for coproducts in the category of abelian groups $\mathbf{Ab}$

I'm working on another problem of Aluffi's Algebra. Given the category $\mathbf{Ab}$ of abelian groups, the problem is to show that for any two groups $G$ and $H$ the product group $G\times H$ ...
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0answers
35 views

Automorphisms of $Z_{p^{i_1}}*Z_{p^{i_2}}*…*Z_{p^{i_n}}$

If $Z_{p^{i_1}}\times Z_{p^{i_2}}\times\cdots\times Z_{p^{i_n}}=\langle a_1,...,a_n\rangle$, then each automorphism of this group is the forms as follows, $$\sigma:a_j\rightarrow ...
4
votes
1answer
42 views

Existence of projectives in the category of torsion abelian groups

Consider the category of torsion abelian groups. This category doesn't have enough projectives by the following argument. Suppose $C_2$ (cyclic group of order 2) is the homomorphic image of a ...
0
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1answer
34 views

Rubik's Slide Proof's and Symmetries in a Rubik's Slide

$\quad$In the February edition of The Mathematical Association of America Monthly there is a article called "$\mathit{Rubik's\ on\ the\ Torus}$". Where they are dealing with solving problems involving ...
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1answer
50 views

Isomorphisms based on Conjugacy Classes

For what groups of the same order are not isomorphic and contain the same conjugacy class? I as well have a more detailed question: For which of those groups are not abelian. The only example I know ...
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2answers
43 views

Show that $G' = \bigcap_{C\subseteq N \triangleleft G} N$

I am asked to show that $G' = \bigcap_{C\subseteq N \triangleleft G} N$, where $G'$ is the commutator subgroup of $G$, and $C :=\{aba^{-1}b^{-1}\mid a,b\in G\}$. Showing $\bigcap_{C\subseteq N ...
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1answer
28 views

Isomorphisms of LCA Groups

From what I understand, in the category $\mathsf {LCA}$ of lca groups, isomorphisms should respect both topology and group structure, hence they are continuous homomorphisms. I'm trying to learn ...
0
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0answers
40 views

Finding the group structure of a finite ring

Trying to construct an example I built up this finite ring: $$B=\mathbb{Z}/9\mathbb{Z}[x,y,z,w_1,w_2]/(x^3-1,y^3-1,(x-1)(z+3w_1),(y-1)(z+3w_2),w_1^2,w_2^2,z^2)$$ I need to know the structure of ...
3
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2answers
39 views

To prove , if Aut$ (G)$ is trivial then $x^2=e , \forall x \in G$

If for a group $G$ the only automorphism is the identity automorphism , then how do we prove that $x^2=e ,\forall x \in G $ ? I have only been able to prove that $G$ is abelian ; Please Help .
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0answers
35 views

group of order $36$ abelian

Let $G $ be a group of order $36$.How to conclude whether it is abelian or not .I tried using Sylow's theorems by calculating the number of subgroups of order $4$ and $9$ but I am getting so many ...
1
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1answer
25 views

Internal and Direct Product question, need help with explanation

I am asked to express a group G={1,7,17,23,49,55,65,71} under multiplication modulo 96 as an external and internal direct product of cyclic groups. However I also have an example to help with it, but ...
0
votes
1answer
46 views

If Abelian Group $G \cong G^3$, does it follow $G \cong G^2$?

If an Abelian Group $G$ satisfies $G \cong G^{3}$, does it follow that $G \cong G^2$ ? It seems elementary but I can't find it on a standard textbook exercise, and maybe simply because it's false. ...
6
votes
1answer
52 views

If N and every subgroup of N is normal in G then G/N is abelian .

Let $N$ be a normal subgroup of $G$ such that every subgroup of $N$ is normal in $G$ and $C_G(N)\subseteq N $ .Prove that $G/N$ is abelian. I think we need to use that every subgroup of $N$ is ...