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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1answer
72 views

Find the number of elements of order 2 and number of subgroups of index 2.

Let $A= \mathbb Z_{60} \oplus \mathbb Z_{45} \oplus \mathbb Z_{12} \oplus \mathbb Z_{36}$. 1) what is the number of elements in A with order 2. 2) what is the number of subgroups of A ...
14
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2answers
254 views

How to recognize a finitely generated abelian group as a product of cyclic groups.

Let $G$ be the quotient group $G=\mathbb{Z}^5/N$, where $N$ is generated by $(6,0,-3,0,3)$ and $(0,0,8,4,2)$. Recognize $G$ as a product of cyclic groups. Honestly, I do not know how to solve ...
1
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1answer
71 views

Showing that every subgroup of an abelian group is normal [duplicate]

I'm working on a proof to show that every subgroup of an abelian group is also a normal subgroup. Let $G$ be an abelian group and $H$ an arbitrary subgroup of $G$. I want to show that $gHg^{-1} = H$, ...
2
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3answers
61 views

Comparing two quadratic number fields

Let $\;$$K = \Bbb Q$[$\sqrt{-5}$] $\;$and let$\;$ $L = \Bbb Q$[$\sqrt{-6}$]$\,$. While it is clear that, as fields, K and L are distinct, if each arithmetic operation is considered separately, they ...
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1answer
19 views

Order of the group associated to a quotient of a lattice

Let $A=[A_{ij}]$ be a $n\times n$ symmetric positive definite integer-valued matrix. Define elements of $\mathbb{Z}^n$ $v_i=[A_{i1},A_{i2},...,A_{in}]$ where I am treating them as row vectors. ...
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1answer
36 views

Uniform modules and submodules [closed]

A non-zero left module $M$ over a ring with unity is called uniform if any two non-zero submodules have non-empty intersection; $M$ is said to contain enough uniforms if any non-zero submodule ...
2
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0answers
53 views

Find all abelian groups that fit in a given short exact sequence.

I have to find all abelian groups that can appear in this short exact sequence. $0\rightarrow \mathbb{Z} \rightarrow A \rightarrow \mathbb{Z}\oplus\mathbb{Z}_5 \rightarrow 0 $ First of all since ...
4
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1answer
75 views

Why is $\pi_1(F_g)^{ab} = \Bbb Z \langle a_1, b_1, \ldots , a_g, b_g \rangle$?

I am told that for a surface with genus $g$, call it $F_g$, the abelianization of $\pi_1(F_g) = \langle a_1, b_1, \ldots , a_g, b_g \mid [a_1, b_1] \cdots [a_g,b_g] = e \rangle$ is $\pi_1(F_g)^{ab} = ...
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1answer
41 views

Show that $ \mathbb{Z}_6 \oplus \mathbb{Z}_6/ \langle (2,3) \rangle $ is or is not cyclic.

I am asked if $ \mathbb{Z}_6 \oplus \mathbb{Z}_6/ \langle (2,3) \rangle $ is cyclic or not. Work: Well the order of 2 in $\mathbb{Z}_6$ is 3 and the order of 3 in $\mathbb{Z}_6$ is 2. Thus, the ...
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1answer
47 views

Frattini subgroup of a finite elementary abelian $p$-group is trivial

I would like to improve my proof of the following result: If $H$ is a finite, elementary abelian $p$-group, then $\Phi(H) = 1$. Here, $\Phi(H)$ is the Frattini subgroup, defined as the ...
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0answers
23 views

How to prove $G$ is cyclic? [duplicate]

Given that $G$ is a finite abelian group, and for every prime $p$ that divides the order of $G$, there is a unique subgroup of order $p$. How can I prove that $G$ is cyclic?
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1answer
52 views

Find the left cosets of subroups of $S_3$

So I am struggling to understand the definition of a coset. If I have the following symmetric group $S_3=\{1, \sigma, \sigma\tau, \sigma\tau^2, \tau, \tau^2\}$, where ...
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2answers
39 views

Monomorphism between abelianizated groups

I want to find an example of a group monomorphism, $$\begin{matrix}\phi:&G_1&\longrightarrow&G_2 \end{matrix}$$ such that, ...
3
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2answers
49 views

$G$ a group s.t. every non-identity element has order 2. If $G$ is finite, prove $|G| = 2^n$ and $G \simeq C_2 \times C_2 \times\cdots\times C_2$

Let $G$ be a group s.t. every non-identity element has order 2. If $G$ is finite, prove $|G| = 2^n$ and $G \simeq C_2 \times C_2 \times\cdots\times C_2$ I know G is abelian since $ab = (ab)^{-1} = ...
2
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1answer
95 views

$G$ is a finite abelian group. For every prime $p$ that divides $|G|$, there is a unique subgroup of order $p$.

$G$ is a finite abelian group. Assume that for every prime $p$ that divides $|G|$, there is a unique subgroup of order $p$. I'd like to prove that $G$ is cyclic. I'm thinking about the approach of ...
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0answers
21 views

A finite group is solvable iff the simple factors in a decomposition sequence are abelian

Show that a finite group $G$ is solvable group (in the sense there exists an $n$ such that $G^{(n)}=1$) if and only the simples factors in a decomposition sequence of $G$ are all abelians. I'm not ...
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1answer
95 views

Every group with 5 elements is an abelian group

I have tried to prove that every group with 5 elements is an abelian group using following approach. Is this correct: Note: I do do not want to use Lagranges theorem and I do not know why groups with ...
1
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2answers
36 views

Detail in the proof $\text{rank}(G)=\dim_{\mathbb{Q}}(G\otimes\mathbb{Q})$

Let $G$ be a finitely generated $\mathbb{Z}$-module. I want to show that $\text{rank}(G)=\dim_{\mathbb{Q}}(G\otimes\mathbb{Q})$. I've shown that $G\otimes\mathbb{Q}\cong \mathbb{Q}^{\text{rank}(G)}$ ...
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2answers
93 views

Proof containing abelian groups.

Let $G \leq S_{999}$ be an abelian subgroup of order $|G| = 1111$. Prove that there exists $i \in$ {$1,2,...,999$} such that $\forall α \in G, α(i) = i$. Okay so I came across this problem and even ...
0
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2answers
30 views

Relations on groups.

Let $G$ be a group and $H$ a subgroup of $G$. Define a relation on elements of $G$ by saying that $a \sim b$ if $b^{-1} a \in H$. This relation is : a) reflexive and symmetric, but transitive only ...
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1answer
40 views

Create a base that contains $x$ in a finite generated Abelian group

Let $A$ be a finite generated free Abelian group and $x \in A$ such that $\forall y \in A$ $\forall n>1: x\neq ny $. Prove that there is a base generating $A$ that contains $x$. It seems simple ...
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32 views

Structure of $p$-torsion groups

Let $A$ be an abelian group whose elements have order a power of certain prime $p$, suppose the $p$-torsion elements are finite, must $A$ be of the form $$(\mathbb{Q}_p/\mathbb{Z}_p)^{\oplus r}\oplus ...
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2answers
50 views

Is $h^*$ injective in this case?

Let $N$ and $N'$ be finite rank free $\mathbb Z$-modules. Let $M=\operatorname{Hom}_{\mathbb Z \text{-mod}}(N,\mathbb Z)$ and $M'=\operatorname{Hom}_{\mathbb Z \text{-mod}}(N',\mathbb Z)$ . Suppose ...
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1answer
85 views

Why $G=HK$ and $H\cap K=\{e\} \implies G = H\times K$?

Given that $G$ is finite abelian, in order to show that $G = H\times K$ one only needs to show that $G=HK$ and $H\cap K=\{e\}$. What motivates this and why is it only true for finite abelian groups?
2
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1answer
55 views

The number of nonzero ring homomorphisms $\mathbb Z_{30}\rightarrow \mathbb Z_{42}$ [duplicate]

I have managed to prove the the number of group homomorphisms is $\mathbb Z_m\rightarrow \mathbb Z_n$ is $\gcd (m,n)$, which is my case is $6$. However, I was told that the number of nonzero ...
0
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1answer
35 views

Let $G$ be an Abelian group with $|G| = n$. Let $p$ be prime with $p | n$. Show that the Sylow p-subgroup of $G$ consists of $e$ and ..

Let $G$ be an Abelian group with $|G| = n$ and let $p$ be prime with $p | n$. Show that the Sylow p-subgroup of $G$ consists of $e$ and all elements whose order is a power of $p$. Answer: By ...
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1answer
71 views

Why is $\mathbb{Z}[1/p]$ the direct limit of $\mathbb{Z}\xrightarrow{p}\mathbb{Z}\xrightarrow{p}\mathbb{Z}\to…$?

This is an example from Algebraic Topology, by Hatcher. As far as I understand, I have to take the direct sum of all the $G_i$s (in this case, $\mathbb{Z}\oplus\mathbb{Z}\oplus...$) and quotient out ...
2
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1answer
149 views

Why is Grp not an Abelian Category?

As I understand it, the category of groups (not just abelian groups) satisfies all of the definitions of an abelian category. It has all kernels/cokernels as well as products/coproducts. Further the ...
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32 views

Relation of Smith normal form to basis of subgroup

Let $A$ be a finite abelian group of rank $2$. Let $\left\{ e_{1},e_{2}\right\}$ be a basis of $A$ and let $C=\left\langle 2e_{1}+3e_{2},2e_{1}+6e_{2}\right\rangle $ be a subgroup. (a) Find ...
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1answer
24 views

Exercise 6.79 from Rotman's Advanced Modern Algebra

If $G$ is a nonzero abelian group show that $$\operatorname{Hom}_{\Bbb Z}(G,\frac{\Bbb Q}{\Bbb Z}) \neq \{0\}.$$
2
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1answer
26 views

Group of units of a non-integral quotient ring

I would to like to know which product of cyclic groups the group $A^\times$ of units of the quotient ring $$ A = \mathbb F_5[X] / ((X^2-2)^2) $$ is isomorphic to. I know that $A$ is not an integral ...
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1answer
34 views

Proving that a group is abelian [closed]

Suppose we have a group $G$ with $|G| = 10$. How do I prove that if its center $Z$ is nontrivial, then $G$ is abelian? Thanks.
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2answers
72 views

Isomorphism and $(\mathbb Q,+)$

Prove that $(\mathbb Q,+)$ is not isomorphic to $(H,+) \neq (\mathbb Q,+)$, a proper subgroup of $(\mathbb Q,+)$. $\mathbb Q$ is the rationals. I thought about taking the contradiction ...
2
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1answer
32 views

Open mapping theorem for normed abelian groups

A norm on an abelian group is a function valued in $\mathbb{R}_{\geq 0}$ which satisfies $|x|=0 \Leftrightarrow x=0$, $|{-}x|=|x|$, and $|x+y| \leq |x|+|y|$, not necessarily $|z x| = |z| |x|$ for ...
2
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2answers
47 views

Is there a non abelian group that characterize a one dimensional lattice structure?

Of all groups that characterize a one dimensional lattice structure (symmetry operations including translation, $C_2$, mirror plane, inversion point), is there a non abelian one? Moreover, Can it have ...
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1answer
51 views

Abelian Group G and it's order (Let G be an abelian group, and let a∈G)

I already asked this question. However, it was closed off. I now have inserted by attempt at the solution: Let $G$ be an abelian group, and let $a\in G$. For $n≥1$, let $G[n:a] := \{x\in G:x^n ...
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0answers
34 views

Abelian Groups and orders

I came across this question, and was wondering what the notation G[n : a] means. Is x the generator? Any comments/explanations will be of great help. The question I found was: Let $G$ be an abelian ...
10
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2answers
114 views

Does there exist an $n$ such that all groups of order $n$ are Abelian?

I know that all groups of order $\leq$ 5 are Abelian and all groups of prime order are Abelian. Are there any other examples? If so is there something special about the orders of these groups?
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2answers
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Struggling to understand example of Ideal which is not finitely generated

I'm working through an algebraic number theory book, but I can't understand the example shown below: I follow the example up till it assumes that $\frac{p_1}{q_1},...,\frac{p_n}{q_n}$ are the ...
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3answers
76 views

Two term free resolution of an abelian group.

This is probably a very easy question but I think I am missing some background regarding free abelian groups to answer it for myself. In Hatcher's Algebraic Topology, the idea of a free resolution is ...
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0answers
42 views

Difficulty understanding subgroups for certain simple groups

Firstly, I am studying at the high-school level so please excuse my lack of understanding of these concepts. Consider the group $\Bbb C_{3v}$ of symmetries of an equilateral triangular lamina. It ...
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0answers
42 views

Extending a homomorphism from a subgroup to whole group where the target is not a divisible group

I was reading this post of stack exchange. So in the question if the circle group is replaced by $\mu_{p-1}$ which is the group of $(p-1)^{th}$ root of unity and if the group $G/H$ is assumed to a ...
2
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1answer
40 views

Number of group homomorphism from $Z_8$ ⊕ $Z_2$ to $Z_4$ ⊕ $Z_4$

I know that there does not exist an isomorphism from $Z_8$ ⊕ $Z_2$ to $Z_4$ ⊕ $Z_4$ as there exist an element of order 8 in $Z_8$ ⊕ $Z_2$ and no element of order 8 in $Z_4$ ⊕ $Z_4$. But what about ...
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1answer
48 views

Proving that a homomorphism is abelian [duplicate]

The question states... Let $\phi: G \to G$ be a homomorphism with the map $\phi(g)=g^2$. Prove $\phi$ is abelian. So far I have: Let $g$ and $h$ be in $G$. Then $\phi(g)=g^2$ and $\phi(h)=h^2$. ...
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33 views

Order of elements in certain generating sets of non-abelian groups !!!

The following example is just to clarify the idea. Example: The dihedral group has the following presentation $$D_{2i}=\left<s,r/s^2=r^i=e,sr=r^{-1}s \right>$$ Let $S_1=\{s,r\}$, ...
0
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1answer
54 views

Subgroups of abelian-by-finite groups

I am trying to prove that a subgroup of a abelian-by-finite group is also abelian-by-finite. I am not sure if I can use the same procedure that is used for a subgroup of a ...
1
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1answer
44 views

Divisible group $G\neq 0$ is not free

How do I show that a divisible group $G\neq 0$ is not free? I know that divisible means that for all elements $g$ in an abelian group $G$ and $n\in\mathbb{N}$ there is an element $a\in G$ such that ...
3
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0answers
54 views

Structure theorems for infinitely generated Abelian groups

The classification theorem for finitely generated Abelian groups is well known and plays big role in mathematics. Are there any structure theorems about infinitely generated Abelian groups known?
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1answer
69 views

Isomorphism between $A$ and $B$ if $A/kA$ and $B/kB$ have same order? [closed]

Let $A$ and $B$ be finitely generated abelian groups, where for all $k\geq1$ the orders of $A/kA$ and $B/kB$ are equal. Are $A$ and $B$ necessarily isomorphic? Why (not)?
0
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1answer
99 views

Surjective endomorphism of abelian group is isomorphism

Let $A$ be a finitely generated abelian group and $f:A\rightarrow A$ a surjective homomorphism. How do I prove that $f$ is an isomorphism? And if $f$ were injective instead of surjective would the ...