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
347 views

Free abelian group $F$ has a subgroup of index $n$?

Suppose that we have a free abelian group $F$. How can it be proved that $F$ has a subgroup of index $n$ which $n≥1$? Honestly, according to the Theorems, I just know that if we take $X$ as a base ...
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1answer
1k views

Is it true that $\mathbb{R}$ and $\mathbb{R}^2$ are isomorphic as abelian groups?

I think the answer is yes. Sketch of the proof Consider $\mathbb{R}$ as a vector space over $\mathbb{Q}$. Let $\{e_\lambda:\lambda\in\Lambda\}\subset\mathbb{R}$ be its Hamel basis. Then ...
2
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1answer
640 views

Determining the Smith Normal Form

Consider the integral matrix $$R = \left(\begin{matrix} 2 & 4 & 6 & -8 \\ 1 & 3 & 2 & -1 \\ 1 & 1 & 4 & -1 \\ 1 & 1 & 2 & 5 ...
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1answer
1k views

An abelian group of order 100

The first part of the problem asks you to prove that an abelian group $G$ with order $100$ must contain an element of order $10$. For this part, I use Sylow theorem to list possiblities for $H$ and ...
6
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1answer
2k views

Converse of Lagrange's theorem for abelian groups

I'm trying to prove that the converse of Lagrange's theorem is true for finite abelian groups (i.e. "given an abelian group $G$ of order $m$, for all positive divisors $n$ of $m$, $G$ has a subgroup ...
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5answers
230 views

How to prove a group has a basis with exactly one element?

I am struggling with the following question. Suppose I have a group $H$ which is a subgroup of $\mathbb{Z}\oplus\mathbb{Z}$, such that any element $\begin{bmatrix} a \\[0.3em] b ...
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1answer
90 views

exact sequence of finite abelian groups which are squares

Let $0 \to A \to B \to C \to 0$ be an exact sequence of finite abelian groups. Assume that $B$ and $C$ is a square (i.e. there are groups $D,E$ such that $B \cong D^2$, $C \cong E^2$). Does this imply ...
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2answers
333 views

Classify finitely generated modules over the ring $\mathbb{C}[\epsilon]$ where $\epsilon^2=0$

Classify finitely generated modules over the ring $\mathbb{C}[\epsilon]$ where $\epsilon^2=0$ Since $\mathbb{C}[x]$ s noetherian we have that $\mathbb{C}[x]/(x^2)$ is too. And thus finitely generated ...
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1answer
109 views

Finding subgroups of index 2 of $G = \prod\limits_{i=1}^\infty \mathbb{Z}_n$

I looked at this question and its answer. The answer uses the fact that every vector space has a basis, so there are uncountable subgroups of index 2 if $n=p$ where $p$ is prime. Are there ...
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2answers
113 views

Torsion subgroup quotient

Let G be an abelian group, $T$ the torsion subgroup of $G$. If $G/T$ is torsion-free, then $T$ and $G/T$ must be disjoint. $G=T \bigoplus G/T$ implies this as well. I don't understand why they are ...
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1answer
115 views

Trouble with decomposition of groups of order 2009

A question says: prove or prove or disprove that there are only 2 non-isomorphic abelian groups of order 2009. I think that it is true because... I split up $2009$ into $7 \times 7 \times 41$ and so ...
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2answers
2k views

Proving that a subgroup of a finitely generated abelian group is finitely generated

A question says: Using the isomorphism theorems or otherwise, prove that a subgroup of a finitely generated abelian group is finitely generated. I would say that for a finitely generated abelian ...
3
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1answer
210 views

Finding subgroups of index 2

Let $G = \prod_{i=1}^\infty \mathbb{Z}_2$ with addition mod 2. I am trying to find subgroups of index 2. I see that taking the entire space and removing all sequences which have a 1 in a certain ...
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3answers
1k views

A nonsplit short exact sequence of abelian groups with $B \cong A \oplus C$

A homework problem asked to find a short exact sequence of abelian groups $$0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$$ such that $$B \cong A \oplus C$$ although the sequence does ...
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1answer
285 views

how to show that a group is elementarily equivalent to the additive group of integers

Is there any fairly easy way of showing a group is elementarily equivalent to the additive group of the integers? I've found a simple characterization here: A ‘natural’ theory without a prime model, ...
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1answer
174 views

Simplify the category of finite abelian groups

Consider the category $\mathsf{FinAb}$ of finite abelian groups. The structure theorem tells us that we can write down a skeleton for this category (a set of representatives for the isomorphism ...
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1answer
90 views

What are the Subgroups of the abelian group $\mathbb{Z}_5$?

How can I find the subgroups of abelian group $\mathbb{Z}_5$? From Lagrange's theorem, the size of the subgroup should divide 5 in this case. So the size of the subgroup should be 1 or 5 ...
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0answers
71 views

Normal and Abelian groups? [duplicate]

Possible Duplicate: A, B subgroups of G, B/A abelian. Show that BN/AN is abelian. Let A, B, N be subgroups of a group G such that A $\triangleleft$ B and B/A is Abelian. Also suppose N ...
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2answers
609 views

How can I prove an abelian group is not free?

How can I prove a given abelian group; such as $\mathbb{Z}_4$ with addition mod 4, is not a free group? Should I consider all the subsets of the given group and prove any of them cannot be a basis? ...
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1answer
388 views

Show that fiber products exist in the category of abelian groups.

Show that fiber products exist in the category of abelian groups. In fact, If $X, Y$ are abelian groups with homomorphisms $f: X \to Z$ and $g: Y \to Z$ show that $X \times_z Y$ is the set of all ...
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1answer
4k views

Computing the Smith Normal Form

This question is related to the Smith Normal Form of Matrices: Let $A_R$ be the finitely generated abelian group, determined by the relation-matrix $R :=$ $$ \begin{bmatrix} -6 & 111 & ...
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2answers
842 views

If a group is $3$-abelian and $5$-abelian, then it is abelian

In a group $(Z,*)$, $(a*b)^{5}=a^{5}*b^{5},\forall a,b\in Z$ and $(a*b)^{3}=a^{3}*b^{3}$ then prove that $Z$ is abelian. I know that for three consecutive integer if $(a*b)^{i}=a^{i}*b^{i},\forall ...
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1answer
152 views

Dimension of subspace fixed by subgroup representation.

If $G$ is an abelian group with cyclic subgroup $H$ and $(\rho,V)$ is a (permutation) representation of $G$. Then I can form a representation of $H$ by considering the composition ...
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2answers
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Show that a group of order $5$ is always abelian.

Show that a group of order $5$ is always abelian. I know that if the binary operation $*$ is defined on a group $G$ is commutative (i.e. $a*b=b*a\ \forall a,b\in G$),\ then G is called a ...
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1answer
203 views

If $G$ has a normal subgroup of order 2 and infinite cyclic quotient, $G$ is abelian?

Assume that $G$ has a normal subgroup $H$ of order $2$ (isomorphic to $Z_{2}$) and $G/H$ is infinite cyclic (which indicates that $G$ is also infinite order). The target here is to prove that $G$ is ...
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1answer
330 views

abelian transitive subgroups

Can anybody tell me what is known about the classification of abelian transitive groups of the symmetric groups? For instance: Let $G$ be a an abelian transitive subgroup of the symmetric group ...
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1answer
148 views

Find the structure of $\mathbb{Z}[\sqrt[3]{2}]/(4+\sqrt[3]{4})$

Let $A=\mathbb{Z}[\sqrt[3]{2}]$ and $I=(4+\sqrt[3]{2^2})$. Elements in $A$ have the form $a\cdot 1+b\cdot 2^{\frac{1}{3}}+c\cdot 2^{\frac{2}{3}} \Rightarrow$ elements in $I$ have the form $$ (a\cdot ...
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1answer
106 views

Finitely generated abelian group has a regular normal form

Prove that every finitely generated abelian group admits a regular normal form. I am having some trouble getting my head wrapped around this problem. If anyone can offer suggestions or help it would ...
3
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3answers
186 views

Orthogonality relations of Characters

Could somebody please help me understand the jump from Proposition 10 to Proposition 11 in the following http://www.ms.uky.edu/~pkoester/research/charactersums.pdf Note: The orthogonality relations ...
0
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3answers
176 views

Prove abelian group

I am given $((0,1),*)$ Where $x,y\in (0,1)$ and $*$ is defined as $x*y=\frac{xy}{1-x-y+2xy}$ How should I go about finding the inverse and identity elements?
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3answers
684 views

On the Factor group $\Bbb Q/\Bbb Z$ [duplicate]

Possible Duplicate: $\mathbb{Q}/\mathbb{Z}$ has a unique subgroup of order $n$ for any positive integer $n$? I have the factor group $\Bbb Q/\Bbb Z$, where $\Bbb Q$ is group of rational ...
3
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1answer
91 views

Expression of an abelian group

Let $A$ be a abelian group generated by elements $\langle a_1,a_2,a_3\rangle$ and $B$ be a subgroup generated by $\langle b_1,b_2,b_3\rangle$ where $\begin{pmatrix} b_1\\ b_2\\b_3 \end{pmatrix}= ...
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2answers
237 views

Canonical form of an Abelian group

Given the abelian group : $A=\mathbb{Z}_{36} ×\mathbb{Z}_{96}×\mathbb{Z}_{108}$ I need to write the canonical form of $18A$ and $A / 18A$ Here is my calculation ,using the followings: ...
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1answer
122 views

Classifying some abelian groups of order $2^5\times 3^5$

I'm requested to classify the abelian groups $A$ of order $2^5 \times 3^5 $ where : $| A/A^4 | = 2^4 $ $ |A/A^3 | = 3^4 $ I need to write down the canonical form of each group . My question is, ...
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2answers
522 views

Factor groups and Torsion subgroups

If $G$ is abelian, $T(G)$ is the torsion subgroup, then $G/T(G)$ is torsion free. If $T(G) = \{1\}$, then $G$ is called a torsion-free group. Below is what I did to prove this statement. ...
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4answers
1k views

How many non-isomorphic abelian groups of order $\kappa$ are there for $\kappa$ infinite?

Let $\kappa$ be an infinite cardinal. How many non-isomorphic abelian groups of order (cardinality) $\kappa$ are there? For finite $\kappa,$ we can use the classification theorem and obtain the ...
3
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1answer
399 views

Quotient Group Classification

I needed help in classifying the following quotient groups according to the fundamental theorem of finitely generated abelian groups: $$ \begin{array} &(\mathbb Z_4 \times \mathbb ...
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1answer
2k views

Every cyclic group $G$ is abelian; is every abelian group $G$ cyclic?

Given the Euler group $U_8= \{1,3,5,7\}$ , I wanted to check the followings : if it is Abelian if it is Cyclic Let's check: Abelian : each $x,y \in G$ : $xy=yx$ , hence indeed abelian. ...
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1answer
418 views

Representation theory/Finitely generated abelian groups

This is not homework. Motivation : I have been reading "Representation and characters of groups" by James & Liebeck, and in chapter 9 they introduce Schur's Lemma, which states that for a finite ...
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3answers
374 views

$U_{14}$ and $U_{18}$ : why are they isomorphic to $\mathbb{Z}_2 \times \mathbb{Z}_3$?

I've tried to check if $U_{14}$ and $U_{18}$ are isomorphic to each other , so first : $U_{18}=\{1,5,7,11,13,17\}$ $U_{14}=\{1,3,5,9,11,13\}$ Both of them of order 6, cyclic, and probably abelian ...
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1answer
875 views

Subgroups of a finitely generated abelian group without torsion

If $G\cong \mathbb{Z}\times \mathbb{Z}\times \dots \times \mathbb{Z}$ is a finitely generated abelian group without torsion of rank $n$, where $n$ is the number of copies of $\mathbb{Z}$. Then any ...
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2answers
695 views

How to show that these two groups are isomorphic?

Let $A$ be a finite abelian group and $p$ be a prime, $p$ divides the order of $A$. Define: $A^p=\{a^p | a\in{A}\}$ and $A_p=\{x\in{A}|x^p=1\}$, where $1$ is the identity in $A$. Show that ...
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1answer
811 views

Find the number of subgroups of order $p$

Let $G$ be a finite abelian group of order $p^n$, where $p$ is a prime number. How to find the number of subgroups of order $p$? i.e. find a formula for the number of subgroups of order $p$. I know ...
2
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1answer
157 views

Why is $\mathbb{F}_q^{*}/(\mathbb{F}_q^{*})^n\rightarrow \boldsymbol{\mu}_n: \overline{x}\mapsto x^{(q-1)/n} $ a group isomorphism?

Let $\mathbb{F}_q$ denote a finite field. Let $n\geq 1$ be an integer such that $n\mid q-1$. Hence the $n$-th roots of unity $\boldsymbol{\mu}_n$ are contained in $\mathbb{F}_q$. Why is the map ...
2
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0answers
142 views

Classification Theorem Of Abelian Groups-Question regarding the proof

I'm currently reading Munkres-Algebraic topology text, and in his review chapter of abelian groups, he gives the classification theorem for finitely generated abelian groups. He ommited some important ...
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1answer
120 views

Uncountably many subgroups of an abelian group

Does the abelian group $\mathbb{Z}[\frac{1}{2}]$ have uncountably many subgroups?
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2answers
514 views

Structure theorem for finitely generated abelian groups

How can we use fundamental theorem of finitely generated abelian groups to list all abelian groups of order 16 up to isomorphism.
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1answer
55 views

If E is a subset of a lattice closed under addition then is the intersection of E with the opposite of some translate finite?

this seems intuitive to me but I'm struggling to prove it (is it false?). Let $E$ be a subset of a lattice (free abelian group of finite rank) closed under addition, containing the origin and such ...
7
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2answers
403 views

Why do torsion-free abelian groups admit linear orders?

I have read a theorem that says that every torsion-free abelian group admits a linear order. The proof used tensor products and so was above my head. I tried to find another proof on the web and I ...
3
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1answer
572 views

Computing index of a subgroup of a free abelian group

We looked briefly at this example in class but I'm not quite sure how to proceed, and I can't find examples of this in any textbooks I have (Dummit & Foote and Nicholson). Suppose we have $H = ...