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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Pisier's $\epsilon$-net condition

I'm reading a book about Sidon sets and I'm stuck on the following proof. In order to facilitate the comprehension of my problems I will give the full proof and the context. Let $G$ be a compact ...
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2answers
86 views

For any element $g$ of $G,$ where $g$ has order $2,$ define $gH=\{gh│h∈H\}$. Prove that the set $K=H∪gH$ is a subgroup of $G.$

Does this solution make sense? Let $G$ be an abelian group and $H$ a subgroup. For any element $g$ of $G,$ where $g$ has order $2$, define $gH=\{gh│h∈H\}$. Prove that the set $K=H∪gH$ is a ...
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1answer
36 views

Finding a subgroup of an abelian group that is isomorphic to Z

The question: If G is an abelian group and f is a surjective homomorphism from G to Z with kernel K, prove that G has a subgroup H such that H is isomorphic to Z. By the first isomorphism theorem I ...
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1answer
38 views

Relation between finite Abelian Groups and traces?

I have recently read Kronecker's 1870 paper on finite Abelian groups, on the definition of abstract group and so on. It turns out that such definition is literally taken over (being probably unaware ...
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1answer
48 views

Algebraic groups?

I have been doing group theory lately but I can not seem to find what I am looking for online (partly because I am not entirely sure what I am looking for). An example of one of the questions: If ...
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1answer
47 views

Example of Abelian Group of order 2014 [closed]

What are some examples of Abelian Groups of order $2014$ ?
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1answer
32 views

Pontryagin Dual of the Unit Circle

Define the unit circle as $\frac{\mathbb{R}}{2\pi\mathbb{Z}}.$ I know the Pontryagin dual (looking at properties of the Fourier transform on locally compact Abelian groups) is $\mathbb{Z}$ but why? ...
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32 views

Invariant factors of a subgroup of a subgroup of $\mathbb{Z}^2$

Consider groups $B\leq A\leq\mathbb{Z}^2$. We have: A basis $e_1,e_2$ of $\mathbb{Z}^2$ and integers $a_1,a_2$ such that $a_1e_1,a_2e_2$ is a basis for $A$, and $a_1\mid a_2$. A basis $f_1,f_2$ of ...
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35 views

Direct sum of Abelian groups and Isomorphism

I'm currently reviewing my algebra for my last prelim and came across the following problem that has me stumped: If $A,B,C $ are finite Abelian groups such that $A\oplus B \cong A\oplus C$ then show ...
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8 views

Maximal subgroups of elementary abelian p-groups

Please how can we characterise the maximal subgroups of a given elementary abelian p-group? How many maximal subgroups does the elementary abelian 2-group of rank n have? What are they? How many ...
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1answer
23 views

Homomorphisms preserve solubility of groups, and some others.

Definition: Let $G$ be a group. A subnormal series for G is a chain of subgroups $1 = G_0 \subseteq G_1 \subseteq G_2 \subseteq G_n = G$ such that $G_i$ is normal in $G_{i+1}$ for $i = 0,1, ..., ...
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46 views

If K and H are normal subgroups of $G$, $H \cap K = \{1\}$ and both $G/H$ and $G/K$ are abelian, then $G$ is abelian.

Let G be a group, and $H \trianglelefteq G$, $K \trianglelefteq G$. Prove that if $H \cap K = \{1\}$ and $ G / H $ and $ G/ K $ are abelian, then G is abelian. I've tried to give a proof by ...
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1answer
58 views

Quotient group(Factor group)

Prove that the quotient group $\frac{Z\times Z\times Z}{<(1,1,1)>}$ is an infinite, non-cyclic group. Here Z is the group of integers with operation of addition, $<(1,1,1)$> is the ...
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41 views

Writing an abelian group as a direct sum of cyclic groups

I have this problem Determine an isomorphic direct sum of cyclic groups, where $V$ is an abelian group generated by $x,y,z$ and subject to:: $x+y=0, 2x=0, 4x+2z=0, 4x+2y+2z=0$ So I wrote ...
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2answers
37 views

Is $\mathbb{Z}\times\mathbb{Z}/((6,5),(3,4))$ is finitely generated?

Let $A$ be the quotient of the free abelian group $\mathbb{Z}^2$ by the subgroup generated by $(6,5)$ and $(3,4)$. the question is $A$ is finitely generated? and if Yes. Can we Decompose it into a ...
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1answer
46 views

Characterization of finitely generated $\mathbb{Z}$-modules with the property that each submodule is a direct summand

I want to characterize all finitely generated $\mathbb{Z}$-modules $M$ with the property that each submodule of $M$ is a direct summand of $M$. I think the module has to be torsion but I couldn't say ...
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46 views

An example of a group that that has an order of M that is abelian?

Theory: If G is a finite abelian group, p is prime and p divides the order of G then G has an element of order p. Can anyone think of a counter example for a number n that is not prime, divides the ...
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1answer
18 views

Uniqueness of abelian group structure on a given set and recursive algorithms

If we have some function $f$ under $\mathbb{Z}$ and $$f(a, f(b, c)) = f(f(a, b), c)$$ $$f(a, b) = f(b, a)$$ $$f(a, 0) = a$$ $$f(a, -a) = 0$$ meaning $f$ is an abelian group with an identity element of ...
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1answer
42 views

When is the quotient of two lattices in ${\mathbb Z}^2$ cyclic?

In this question, by a lattice I mean a full-rank subgroup of the group ${\mathbb Z}^2$. What I would like to know is: Can one give a comprehensible description of those lattices ...
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1answer
40 views

Assuming the axiom of choice ,how to prove that every uncountable abelian group must have an uncountable proper subgroup?

Assuming the axiom of choice , how to prove that every uncountable abelian group must have an uncountable proper subgroup ? Related to Does there exist any uncountable group , every proper subgroup ...
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1answer
36 views

Characteristic subgroup of an Abelian-by-Finite Group

Let $G$ be a group such that $A$ is a normal Abelian subgroup and $G/A$ is finite. Is always possible to find an Abelian characteristic subgroup $B$ such that $G/B$ is finite too? Factoring by $G^n$ ...
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43 views

In an abelian group, prove that $x=a_1a_2\cdots a_n$ implies $x\circ x=e$ [closed]

Let $(G,\circ)$ be a group with elements $a_1,a_2,\cdots, a_n$ and $x=a_1a_2\cdots a_n$. Show that $x\circ x=e$.
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Does an injection of finitely generated abelian groups always induce a surjection via $Hom(-,U(1))$?

I was recently interested in the following conjecture, which at first sight seemed pretty elementary. Conjecture: Let $i: A \hookrightarrow B$ be an injection into a finitely generated abelian group. ...
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Why do mathematicians study elementary abelian groups?

I took two algebra courses that I liked as an undergraduate mathematics major in college, but we never covered elementary abelian groups. I recently got interested in the properties of a group I ...
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2answers
79 views

A finite abelian group has order $p^n$, where $p$ is prime, if and only if the order of every element of $G$ is a power of $p$

Suppose that G is a finite Abelian group. Prove that G has order $p^n$, where p is prime, if and only if the order of every element of G is a power of p. I tried the following route, but got stuck. ...
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1answer
66 views

If the commutator subgroup is abelian, is it necessary trivial? [closed]

Let $G$ be a group. We define the commutator of $a$, $b$ in $G$ as $[a,b]:=aba^{-1}b^{-1}$. Let $C=\langle[a,b] \mid a,b\in G \rangle$ be the commutator subgroup of $G$. Suppose that $C$ is abelian. ...
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1answer
36 views

Group ring C[Z/n] and Artin-Wedderburn decomposition

I am trying to answer the following questions, which I assume follow on from eachother each other; Write $\mathbb C$[$\mathbb Z$/n] as a product of simple rings. For abelian groups $G_1$, $G_2$, ...
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1answer
50 views

For a subgroup $H$ of a finite group $G$ , when does $|Aut(H)|$ divides $|Aut(G)|$?

Let $H$ be a subgroup of a finite group $G$, then is it true that $|Aut(H)|$ divides $|Aut(G)|$? What if we also assume $G$ is abelian? (I know that $|Aut(H)| \space \big| \space |Aut(G)|$ if $G$ is ...
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Free and infinite abelian groups and the $\mathbb{Z}$-module structure. [closed]

I want to establish a specific link between the free and infinite Abelian groups and the $\mathbb{Z}$-module structure. Let me explain: 1 - If I have a finite abelian group $A$, is it possible ...
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22 views

Naively showing that $A_n$ mod a nontrivial normal subgroup is abelian.

Suppose $H \lhd A_n$ is a nontrivial normal subgroup of the alternating group on $n$ letters. Without using the fact that $A_n$ is simple, prove that $A_n/H$ is abelian. Can this be done? I will ...
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2answers
60 views

Group Theory: Showing that a subgroup is isomorphic to a product of groups

I have the following question, where the topic being tested is cosets, order and Lagrange's theorem: Suppose that every element $x$ in a group $G$ satisfies $x^2 = e$. Prove that $G $is abelian. ...
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1answer
50 views

How to prove that a certain set is a coset of a subgroup of a group of characters mod $m$?

I have the following question: Let $m>1$ be a positive integer and $G:=G(m):={(\mathbb{Z}/m\mathbb{Z})}^{*}$. Let $p\in\mathbb{P}$ with the property $p\nmid m$. Let $\widehat{G}:=\text{Char}(G)$ ...
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Showing that the group is abelian

Let $\sigma = (123456)$ in $S_6$. And let $G = \{e, \sigma, \sigma^2, \sigma^3, \sigma^4, \sigma^5\}$ be a group under operation from $S_6$. Is $G$ abelian? Workings: A group is abelian if it is ...
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33 views

Why is the group abelian?

Lets say a group $G$ consists of 3 Sylow groups, $H_1,H_2,H_3$. Each of order $p_1,p_2,p_3$, that are prime numbers and different. Since we only have one of each Sylow group for each p, the second ...
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Are there any non-obvious colimits of finite abelian groups?

Does the forgetful functor $U : \mathsf{FinAb} \to \mathsf{Ab}$ from finite abelian groups to abelian groups preserve colimits? Morally this should be true, but it is not so easy (for me) to come up ...
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2answers
63 views

Let G be a group, where $(ab)^3=a^3b^3$ and $ (ab)^5=a^5b^5$. Prove that G is an abelian group. Want to specify.

Let $G$ be a group, where $(ab)^3=a^3b^3$ and $(ab)^5=a^5b^5$. How to prove that $G$ is an abelian group? P.S Why cannot not we just cancel ab out of the middle of these expressions? Why can we only ...
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4answers
107 views

Let $G$ be a group, where $(ab)^3=a^3b^3$ and $(ab)^5=a^5b^5$. How to prove that $G$ is an abelian group?

Let $G$ be a group, where $(ab)^3=a^3b^3$ and $(ab)^5=a^5b^5$. Prove that $G$ is an Abelian group. I know that the answer for this question has been already posted and I have seen it. However, could ...
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1answer
82 views

When the endomorphism ring of an abelian group is generated by automorphisms?

Given an abelian group $M$. First I'd like to know if $\text{End}(M)$ is generated by $\text{Aut}(M)$ (as ring, or equivalently, as additive group). Second I'd like to know if it doesn't hold ...
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1answer
38 views

How to classify all the abelian groups with finite exponent?

Let $A$ be an abelian group, the exponent exp$A$ is the least natural number $n$ (if exists) such that $nA=0$ or $+\infty$. The question can be reduced to the case exp$A=p^n$ for a certain prime ...
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1answer
79 views

Intuition for an abelian fundamental group

Any topological group has an abelian fundamental group by the Eckmann-Hilton argument. Is there some intuition behind the fundamental group being abelian that would enable one to predict this ...
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1answer
44 views

Faithfully Flat Abelian Groups

I need some help to find faithfully flat abelian groups. Flat abelian groups are torsion free $\mathbb{Z}$-modules. But what about faithfully flat abelian groups. $\mathbb{Q}$ is an example that is ...
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1answer
34 views

List all abelian groups that have order 81 and contain an element of order 27

List all abelian groups that have order 81 and contain an element of order 27. For each, give the primary decomposition and a specific element having order 27. I know $81 = 3^{4}$ so the abelian ...
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41 views

Graph Jacobian (Sandpile group) usages

Let $\Gamma$ be a graph (say, finite) and $S_\Gamma$ be it's Jacobian (also known as the sandpile group or Picard group). I'm wondering about what fundamental things one can learn about $\Gamma$ from ...
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2answers
33 views

Give a specific example to show that $\mathbb Z_{2}$ × $S_{4}$ is not abelian.

Give a specific example to show that $\mathbb Z_{2}$ × $S_{4}$ is not abelian. I know that $S_{4}$ is not abelian and therefore $\mathbb Z_{2}$ × $S_{4}$ is not abelian. I'm not sure how to show this ...
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130 views

On subgroups of abelian groups

Let $G$ be a product of $n$ finite cyclic groups. Is every subgroup of $G$ also a product of (at most) $n$ finite cyclic groups ? I do not know the answer to this question, but I'm tempted to say ...
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1answer
42 views

Determining the multiplicative group of a ring of polynomials

Let us say that we have the polynomial ring R[x]. Would it be possible to determine the order of the multiplicative group of R[x] modulo a polynomial f?
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1answer
44 views

If order of group is $p^2$, where $p$ is prime, how can you deduce $G$ is isomorphic to $C_{p^2}$ or $C_p \times C_p$?

Given $|G|=p^2$ then how can you deduce $G\cong C_{p^2}$ or $G \cong C_p \times C_p$ I have shown that G is abelian, not sure what to do next
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1answer
21 views

Given a group is finite and non-abelian, why is the left coset with the centre of the group non-cyclic?

Assume $T$ is finite and non-abelian then why is $T/Z(T)$ non-cyclic? Where $Z(T)$ is the centre of the group $T$. I've shown $Z(T)$ is a normal subgroup of T, but not sure what to do next or if ...
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28 views

homorphisms of abelian groups

Please help me with resolving this problem from Romanian "Gazeta Matematica": "a finite Abelian group $G$ such that $|\text{End } G |$ and $|\text{Aut } G |$ are coprime numbers. Show that $G$ is ...
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1answer
20 views

Equivalence of definitions of injective modules

Wikipedia article gives a number of definitions of injective modules, namely: If $Q$ is a submodule of some other left $R$-module $M$, then there exists another submodule $K$ of $M$ such that $M$ is ...