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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2
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1answer
36 views

Perfect pairing induces isomorphism of tensor products

Let $M, N$ be $R$-modules and $(\cdot, \cdot): M \times N \to R$ be a perfect pairing. Wikipedia sais that this means that the map $\varphi: M \to \text{Hom}_R(N, R), m \mapsto (n \mapsto (m, n))$ is ...
2
votes
1answer
28 views

The abelian group of smallest order and smallest non prime integer n divides |G| but G doesn't have an element of order n?

I don't know how to think of an example. What's an example of such a group. It doesn't make sense to me because if it is a finite abelian group, it can be written as a direct product of the integers ...
0
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2answers
69 views

Abelian group which is not one of these

Im struggling to find a finite abelian (commutative , associative) group $(G,\circ)$ with some specific conditions: $a\circ b$ isn't naive addition $a+b$ for $a,b\in G$ $G$ is a subset of ...
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0answers
12 views

I.N. Herstein , Topics in Algebra problem 2.5.24 [duplicate]

Let $G$ be a finite group whose order is not divisible by 3. Suppose that $(ab)^3=a^3b^3$ for all $a,b\in G$. Prove that $G$ must be abelian.
1
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1answer
22 views

How do I check this simple set is an Abelian group?

The n-gon in question is a 3-gon. It is an equilateral triangle to be exact. This is a Dihedral group of order 6 (3 reflections and 3 rotations) I have plotted the Cayley's table. The set of ...
1
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1answer
55 views

Is this true for finite abelian groups?

I'm trying to decompose abelian groups using the structure theorem for modules over PIDs. Here is one I have proved: Let $G$ be a non-trivial abelian group whose prime decomposition of the order ...
-2
votes
2answers
83 views

Is every subgroup of a group normal?

Is there a simple example that can be used to show that not every subgroup of a group is normal? thanks,
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5answers
97 views

Prove every group of order less or equal to five is abelian [closed]

Is it possible to prove that every group of order less or equal to five is abelian? thanks,
0
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0answers
18 views

Looking for an element in a finite abelian group $G$ which is the square of more than two elements of orders $m\geq 3$ in $G$?

Please do we have an element in a finite abelian group $G$ which is the square of more than two elements of orders $m\geq 3$ in $G$? For example in $Q_8$ which is non-abelian, all the elements in ...
1
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2answers
23 views

Locally graded group with all proper subgroups abelian

A group $G$ is said to be locally graded if every finitely generated nontrivial subgroup of $G$ contains a proper subgroup of finite index. I have to prove that a locally graded group with all proper ...
0
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1answer
60 views

if o(a) is equal to exponent of finite abelian group G then $G=<a>\times K$

problem:prove that if $o(a)$ is equal to the exponent of a finite Abelian group $G$, then there exists $H<G$ such that $G=H\times\langle a\rangle$$$$$ using fundamental theorem of finitely ...
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2answers
45 views

What automorphisms exist on the abelian group of positive rationals under multiplication?

Consider the abelian group $(\mathbb{Q}_{>0}, \times)$. What automorphisms exist for this group? I can only think of the trivial one and of $\phi(q) = \frac{1}{q}$. If we relax the problem to ...
3
votes
1answer
82 views

Find the smallest $n$ such that $\mathbb Z_2\times \mathbb Z_2\times \mathbb Z_2$ is isomorphic to a subgroup of $S_n$

Let us consider the group $A=\mathbb Z_2\times \mathbb Z_2\times \mathbb Z_2$. Find the smallest positive integer $n$ such that $A$ is isomorphic to a subgroup of $S_n$. My thought. Since ...
8
votes
1answer
129 views

A problem about abelian group

Given a group $G$, let $G_m$ be the group generated by the set $S=\{g^m|g\in G\}$. Prove that if $G_m$ and $G_n$ are both abelian, then $G_{\gcd(m,n)}$ is also abelian.
1
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1answer
48 views

On a classification of all the characteristic subgroups of a finite abelian $p$-group.

For any finite abelian group $G$, any $n\mid\exp G$ and any $m\mid\frac{\exp G}{n}$, let $nG[m]:=\{g\in nG\mid mg=0\}$. I wonder if every characteristic subgroup of a finite abelian $p$-group $P$ is ...
0
votes
1answer
29 views

Conditions on subgroups $H, K$ of an abelian group $G$ such that $G/K \cong H/(H \cap K)$

I am trying to prove the equivalence of two formulations of simplicial homology on a manifold $X$, both of which are defined as the quotient of a certain set of simplicial chains on $X$ by a certain ...
0
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1answer
42 views

Describing groups with given presentation? $\langle x,y\ |\ xy=yx,x^5=y^3\rangle$ and $\langle x,y\ |\ xy=yx,x^4=y^2\rangle$.

I'm trying to describe the groups with presentations $\langle x,y\ |\ xy=yx,x^5=y^3\rangle$ and $\langle x,y\ |\ xy=yx,x^4=y^2\rangle$. I have some problems getting a good picture of what they look ...
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1answer
25 views

Unique abelian group with 10 elements [closed]

How do you demonstrate that there is an unique abelian group with 10 elements?
4
votes
1answer
56 views

Sum of elements of a finite Field

Let $F$ be a finite field and $i$ an integer. Calculate the sum of all the elements of $F$,each raised to the $i-th$ power. My approach so ...
1
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1answer
25 views

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 ...
3
votes
0answers
99 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 ...
2
votes
3answers
110 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 ...
4
votes
1answer
40 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 ...
0
votes
1answer
42 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 ...
0
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1answer
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 ...
1
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1answer
53 views

Example of Abelian Group of order 2014 [closed]

What are some examples of Abelian Groups of order $2014$ ?
1
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1answer
43 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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0answers
33 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 ...
0
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0answers
39 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 ...
0
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0answers
17 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 ...
1
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1answer
24 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, ..., ...
2
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2answers
47 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 ...
1
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1answer
63 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 ...
0
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0answers
42 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
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 ...
1
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1answer
50 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 ...
0
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3answers
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 ...
1
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1answer
19 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 ...
1
vote
1answer
43 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 ...
1
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1answer
43 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 ...
1
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1answer
45 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 $G^n$ ...
3
votes
1answer
30 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. ...
3
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0answers
64 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 ...
0
votes
2answers
96 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. ...
3
votes
1answer
46 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 $G_1$, $G_2$, ...
3
votes
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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0answers
23 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 ...
0
votes
2answers
66 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. ...
1
vote
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 $\widehat{G}:=\text{Char}(G)$ ...
0
votes
2answers
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 ...