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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Any characterization of $H^2(\mathbb{Z}_n,\mathbb{Z}_m,\theta)$?

I've been reading chapter 7 of An Introduction to the theory of groups by Rotman related to Extensions and Cohomology, and there is something that is not completely clear to me. Given the exact ...
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101 views

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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3answers
120 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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43 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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45 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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49 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
58 views

Example of Abelian Group of order 2014 [closed]

What are some examples of Abelian Groups of order $2014$ ?
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1answer
54 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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34 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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43 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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1answer
27 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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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
70 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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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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42 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
53 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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51 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
21 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
46 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
45 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
47 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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1answer
31 views

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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68 views

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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132 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
52 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
53 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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24 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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68 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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52 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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30 views

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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36 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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123 views

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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68 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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112 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
94 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
41 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
89 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
43 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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1answer
57 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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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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155 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
43 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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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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1answer
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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 ...
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1answer
36 views

Countable LCA groups

Is it true that a countable LCA group can only be discrete ? This question is related to a comment here : A theorem on LCA group - is the uncountability necessary?
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102 views

Example of an infinite abelian group having a non-cyclic finite subgroup [closed]

Give example (if exists) of an infinite abelian group having a non-cyclic finite subgroup . Please help
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1answer
190 views

A group whose automorphism group is cyclic

Is there an Abelian group $A$ which is not locally cyclic whose automorphism group is cyclic ?
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32 views

Abelian group, inverse element

What would be the inverse element for this abelian group $[1,2,3,4,...,p-1]$ , in which p is a prime number, with this operation $(a*b)mod.p$? For all $a$ and $b$ of the set. I know the inverse ...