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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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
46 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
67 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
86 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 ...
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2answers
115 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
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1answer
180 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 ...
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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
91 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
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1answer
65 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 ...
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1answer
77 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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98 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
144 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
61 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 $a_i,b_i\in\Bbb{Z}$. ...
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4answers
173 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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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 $ ...
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12 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 ...
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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
79 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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45 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 ...
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1answer
59 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 : ...
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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 $a,b,c \in \Bbb Z_3$. ...
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2answers
70 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
167 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
442 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
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 ...
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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$
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$G$ 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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48 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 $\mathrm{Cong}(X)$ ...
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2answers
53 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 ...
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1answer
99 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
68 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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34 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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86 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 ...
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0answers
22 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 ...
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1answer
33 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
55 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
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1answer
99 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
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1answer
202 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$ ...
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1answer
63 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
26 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 $G$ ?
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1answer
66 views

Automorphisms of the Prüfer 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
47 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 $(S,*)$ ...
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1answer
21 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 ...
10
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1answer
214 views

Writing $G/A\times G/B$ 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
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1answer
84 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
53 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
38 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
63 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
38 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
55 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 ...