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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A question about tensor product of finite abelian groups

If $G$ and $H$ are finite abelian groups, then $G \otimes_\mathbb{Z} H \cong Hom_\mathbb{Z}(G,H)$ ?
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1answer
14 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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21 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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30 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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5 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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2answers
44 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
53 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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1answer
36 views

characterization of some finitely generated Z-modules

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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2answers
35 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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40 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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3answers
44 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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98 views

A question about the automorphism group of $\mathbb{Z}_{2} \times \mathbb{Z}_{4}$

I wanted to clarify some confusion I was having on the automorphism group of $\mathbb{Z}_{2} \times \mathbb{Z}_{4}$, which I call $Aut(\mathbb{Z}_{2} \times \mathbb{Z}_{4})$. I considered the ...
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0answers
57 views

Number of group homomorphisms between two finite groups

I am confused between the Answer of this Question 1 and the Answer of this Question 2. In answer of the 1st question groups should must be Abelian whether in the answer of 2nd question there are no ...
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1answer
39 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
38 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
31 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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2answers
42 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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1answer
27 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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0answers
62 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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2answers
70 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
65 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
33 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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2answers
17 views

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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0answers
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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3answers
406 views

If $|\lbrace g \in G: \pi (g)=g^{-1} \rbrace|>\frac{3|G|}{4}$, then $G$ is an abelian group.

Assume that $\pi$ is an automorphism of a finite group $G$. Let $S$ denote the set $\lbrace g \in G: \pi (g)=g^{-1} \rbrace$. Show that if $|S|>\frac{3|G|}{4}$, then $G$ is an abelian group. ...
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2answers
54 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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4answers
215 views

Let $G=\mathbb Z_{10}\times\mathbb Z_{15}.$ How many elements of given orders? [closed]

Let $G=\mathbb Z_{10}\times\mathbb Z_{15}.$ Then which of the followings are correct: $G$ contains exactly one element of order $2;$ $G$ contains exactly $5$ element of order $3;$ $G$ contains ...
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1answer
49 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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1answer
43 views

Number of elements in a group

The group $G$ consists of the binary strings of length $5$ under addition $\mod 2$ in each component. (It is isomorphic to $(\mathbb Z_2)^5$, the direct product of $5$ copies of $\mathbb Z_2$.) I ...
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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 ...
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1answer
32 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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1answer
136 views

CEP for Abelian groups and lattices

An algebra $A$ has the congruence extension property (CEP) if for every $B\le A$ and $\theta \in \operatorname{Con} B$ there is a $\phi \in \operatorname{Con} A$ such that $\theta = \phi \cap (B\times ...
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2answers
193 views

First order theory of abelian groups and first order theory of cyclic groups are coincide?

Let $T$ be a first-order theory of cyclic groups. Even if an abelian group $(G,+)$ satisfy $(G,+)\models T$ there is no reason that $(G,+)$ is a cyclic. (For example, by Löwenheim–Skolem theorem there ...
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1answer
114 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 ...
3
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4answers
105 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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2answers
88 views

Let $A$ be a finitely generated abelian group. Show that $\operatorname{Hom}(A,Z)$ is a free abelian group.

My question is Let $A$ be a finitely generated abelian group. The structure theorem says that $A$ is isomorphic to $F \times T$, where $F$ is isomorphic $\mathbb Z^m$, some $m \geq 0$, and $T $ is ...
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1answer
81 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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2answers
62 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
150 views

Prove that no finite abelian group is divisible.

A nontrivial abelian group $G$ is called divisible if for each $a \in G$ and each nonzero integer $k$ there exists an element $x \in G$ such that $x^k=a$. Prove that no finite abelian group is ...
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1answer
848 views

A finite abelian group G has a subgroup of order d for all d dividing the order of G

Use Cauchy’s Theorem and induction to prove that a finite abelian group $G$ has a subgroup of order $d$ for all $d$ dividing the order of $G$.
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1answer
37 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
78 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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40 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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1answer
43 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
30 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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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 ...
4
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2answers
128 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 ...