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
57 views

find torsion coefficients of groups

I have to find torsion coefficients of groups $G_1\simeq Z/2\oplus Z/4\oplus Z/3\oplus Z/3\oplus Z/9$ and $G_2\simeq Z/15\oplus Z/20\oplus Z/18$. I want to ask if my calculations are correct. For $...
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1answer
25 views

Invariant factors and elementary divisors of an abelian group

I have to find the elementary divisors and invariant factors of : $$ \mathbb Z_6\oplus\mathbb Z_{20}\oplus\mathbb Z_{36}$$ I'm following this. I think that elementary divisors are $\{2,2^2,2^2,3,3^...
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1answer
722 views

Intervals in divisible ordered groups

Is it true that if $(G,+,0,<)$ is a divisible ordered abelian group with at least two elements, then for $a,b >0 \in G$, there is an injective order preserving map from $[0;a)$ to $[0;b)$? It ...
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0answers
23 views

Characters of a Finite Abelian group

Let $C_m$ denote the cylic group of order $m$. Recall $\widehat{C}_m=\{ \psi_j: j\in C_m\},$ where $\psi_j$ is defined by \begin{align} \psi_j(k)=e\Big( \frac{2\pi \sqrt{-1}jk}{m}\Big). \end{align} ...
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1answer
42 views

The central product of two cyclic subgroups of prime power order for one $p$ is isomorphic to direct product of two cyclic groups of prime power order

For two groups $G_1, G_2$ and two central subgroups $U_1 \le Z(G_1), U_2 \le Z(G_2)$ which are isomorphic by some given $\mu : U_1 \to U_2$ the central product is the group $$ (G_1 \times G_2) / D $$ ...
5
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1answer
80 views

Is there a known formula for the “cyclicity” of a positive integer?

Given a positive integer $n$, let us define that the cyclicity of $n$ is the number of multitsets of cyclic numbers (distinct from $1$) whose product is $n$. For example, the cyclicity of $15$ is $2$, ...
2
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1answer
19 views

Z modules spanned by row space of matrix invariant under matrix multiplication

I have met this strange looking problem on which I have no idea, from my course on Abstract Algebra dealing with modules: Let $ v_1,...,v_k \in \mathbb{Z}^n $ row vectors of length n over $ \...
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4answers
148 views

Why are there $12$ automorphisms of $\Bbb Z\oplus \Bbb Z_{3}$?

Let $A:=\Bbb Z\oplus \Bbb Z_{3}$, then what is $|\text{Aut}(A)|$? My answer is $4$ but the correct answer (without explanation) turns out to be $12$! How come? Well my understanding is, it just ...
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2answers
37 views

A problem in decomposing a p group into direct sum of nontrivial subgroups

Hello all I have taken a group theory course where we are now covering p groups and we I have met the following exercise: Let $ G = Z/(p^n) $ is a(n Abelian) group of order $ p^n $ for a prime $ p ...
5
votes
1answer
83 views

if $ab = ba$ for all $a \in X$ and for all $b \in X$ then $\langle X \rangle$ is abelian subgroup of $G$

if $X \subseteq G$ such that $\forall a,b \in X$ we have $ab = ba$ then we should prove that $\langle X \rangle$ is an abelian subgroup of G. its abviouse that $\langle X \rangle$ is subgroup of $G$. ...
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2answers
60 views

Is every subgroup of a free abelian group a direct summand?

My guess is NO, because take $G=\mathbb{Z}$ and $F=2\mathbb{Z}$ is a subgroup but not a direct summand.
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1answer
32 views

A problem in Abelian p-group being indecomposable

I have recently met this very interesting problem in my Group theory course: Let $ p $ be a prime number and $ 1 \leq n $ is a natural number such that G is the Abelian p-group $ G = Z_{p^n}=Z/(p^...
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2answers
85 views

A finite abelian group $A$ is cyclic iff for each $n \in \Bbb{N}$, $\#\{a \in A : na = 0\}\le n$

Let $A$ be a finite abelian group. Prove that $A$ is cyclic iff for each $n \in \Bbb{N}$ $$\#\{a \in A : na = 0\}\le n.$$ Any help or hint will be appreciated.
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2answers
73 views

Examples of torsion-free abelian groups with finite automorphism group

$\mathbb{Z}$ is a torsion-free abelian group with finite automorphism group. Are there other examples of such groups? Jumping from $\mathbb{Z}$ to $\mathbb{Q}$ is not good; since $\mathbb{Q}$ has ...
3
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1answer
53 views

Minimal number of generators for a finitely generated abelian $p$-group

Let $A = \text{Tor}_p(A)$ be a finitely generated abelian $p$-group. (Here $p$ is prime). Show that the minimal number of generators of $A$ is $\log_p|A/pA|$. What I tried - I think that from the ...
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2answers
75 views

Show that $\mathbb Z_p \oplus \mathbb Z_p \oplus \mathbb Z_p$ is not generated by two elements

Show that group $A = \mathbb Z_p \oplus \mathbb Z_p \oplus \mathbb Z_p$ is not generated by two elements. ($p$ is prime.) any help or hint will be appreciated.
2
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0answers
49 views

Number of constituents in invariant factor decomposition of kernel of homomorphism

Notation. Given a finite abelian group $ G $, the invariant factor decomposition theorem ensures a the existence of $ k_1 \mid \cdots \mid k_n $, all different, such that $ G \simeq \bigoplus_{i=1}^n \...
5
votes
1answer
41 views

Example of an endomorphism on an abelian group that is not left multiplication

It is well-known that all endomorphisms on the abelian group ($\Bbb{Z}$,+) can be seen as a left multiplication by some element in some ring structure on ($\Bbb{Z}$,+); namely left multiplication by ...
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4answers
104 views

Is there a non abelian group of order 759? [closed]

I tried to use Sylow theorems to prove that there is not, but it is not trivial.
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3answers
72 views

Number of elements of order $2$ in Abelian groups of order $2^{n}$

I'm self studying some group theory and one of the exercises I came across: Question: Prove that an Abelian group of order $2^{n}, n \in \mathbb{N}$ must have an odd number of elements of order 2. ...
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1answer
86 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 with ...
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2answers
272 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 these ...
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1answer
82 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 $gHg^{-1} = H$, ...
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votes
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
25 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
48 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
56 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
76 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} = \...
0
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1answer
42 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 ...
1
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1answer
64 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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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 $\sigma$=$\left(\begin{array}{ccc}...
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2answers
42 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, $$\begin{matrix}\bar\phi:&Ab(G_1)&\longrightarrow&Ab(...
3
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2answers
54 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} = b^{...
2
votes
1answer
100 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 ...
0
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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 ...
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2answers
39 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
94 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 ...
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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 ...
2
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1answer
41 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 ...
2
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0answers
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
88 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?
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1answer
58 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 (non-...
0
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1answer
36 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 Sylow 1,...
4
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1answer
77 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
167 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 ...
0
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0answers
34 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 a ...
1
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1answer
26 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
32 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 ...