Questions tagged [abelian-groups]

For questions about abelian groups, including the basic theory of abelian groups as a topic in elementary group theory as well as more advanced topics (classification, structure theory, theory of $\mathbb{Z}$-modules as related to modules over other rings, homological algebra of abelian groups, etc.). Consider also using the tag (group-theory) or (modules) depending on the perspective of your question.

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When all commutant (centralizer) subgroups are abelian

I have seen the following problem in chapter 9 of Abstract Algebra by Dan Saracino: Let $G$ be a group and for $a,b \in G$ let $a\ R\ b$ mean that $ab=ba$. Must $R$ be an equivalence relation on $G$? ...
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Suppose $G$ is abelian and finite of order $n$ and $x\in G$. Prove that the order $x$ divides $n$. [duplicate]

Suppose $G$ is abelian and finite of order $n$ and $x\in G$. Prove that the order $x$ divides $n$. My solution: Denote $m:=\text{the order of }x.$ Suppose the order of $x$ does not divide $n$. Then $$...
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3 votes
1 answer
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Conjugacy classes of quaternion groups $Q_n=\langle x,y\ |\ x^{2n}=1,\ x^n=y^2,\ yxy^{-1}=x^{-1}\rangle$

The conjugacy classes of the quaternion group $Q_2=\langle i,i\ |\ i^4=j^4=1,\ ij(-i)=-j\rangle$ are $\{1\},\ \{-1\}, \{i,-i\}, \{j,-j\}$ and $\{ij,-ij\}.$ Proceding in a similar manner I am trying ...
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Is $G\otimes\mathbb Z^X$ isomorphic to $G^X$?

Let $G$ be an abelian group and let $X$ be a finite set. For a given group $T$, let $T^X$ be the group of functions $X\to T$ where the operation is defined pointwise. Is it true that $G\otimes\mathbb ...
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Number of generators of nontrivial finite abelian group

(Dummit Foote Exercise 5.2.11) Let $G$ be a nontrivial finite abelian group of rank $t$ (i.e., $G\simeq\Bbb Z/n_1\times\cdots\times\Bbb Z/n_t$ such that $n_{i+1}\mid n_i$ for $1\leq i\leq t-1$). (a) ...
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5 votes
1 answer
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Must every group with a one-half morphism embed into a ring in which the morphism is division by 2?

Let $\alpha$ be an endomorphism of a group $G$. We say $\alpha$ is a one-half morphism if $\alpha(gg) = g$ for all $g \in G$. The existence of such a morphism in fact implies that $G$ is abelian, ...
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8 votes
1 answer
136 views

Is every ring a homomorphic image of some abelian group's endomorphism ring?

Question: Is every ring a homomorphic image of some abelian group's endomorphism ring? I'm a beginner in ring theory. I don't know if this is true and google didn't really help me out here.
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In order for the sizes of subgroups of multiplicative group to be equal, the subgroups must be equal [duplicate]

I'm having trouble explaining this problem. Proof or refute the following statement: Let H and K be subgroups of group Z#n (Multiplicative group). Then |H| = |K| if and only if H = K Thanks for help
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Abelian groups for which every direct sum decomposition contains a Boolean summand

I am interested in the following Question Let $G$ be a countable abelian group with the property that whenever $G\cong H\times K$, at least one of the factor is Boolean. Is it always true that $G\cong ...
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1 answer
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Set of homomorphisms on a free abelian group is a free abelian group.

If $G$ is a free abelian group with rank $n$, I need to show that ${\rm Hom}(G,\mathbb{Z})$, set of all homomorphisms is also free abelian group of rank $n$, My work: Since $G$ is free abelian group ...
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3 votes
0 answers
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Dummit and Foote $2.4.20$, divisible group

Q. Prove that if $A$ and $B$ are nontrivial abelian groups, then $A \times B$ is divisible if and only if both $A$ and $B$ are divisible groups. My approach Let $A, B$ be two divisible groups, and ...
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Abelian group decomposition

I'm trying to describe equivalent classes of canonical decompositions of abelian groups(canonical decompositions are equivalent if and only if they are differing by rearrangement of terms) It's easy ...
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2 votes
2 answers
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Classifying abelian groups of order $56$.

This is an exercise from Dummit and Foote which asks me to show that the number of abelian groups of order $56$ are $3$ upto isomorphism However I am getting $4$. I am not able to understand which one ...
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49 views

Let $G$ be a group with $K\unlhd G$ and Abelian $N\unlhd G$. Is $G/(NK)$ an Abelian group?

Let $G$ be a group such that $N$ is an Abelian normal subgroup of $G$ and $K$ is just a normal subgroup of $G$, is $G/(NK)$ an Abelian group? Also we know that $NK$ is normal in $G$. I am not really ...
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1 vote
1 answer
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Show that the free abelian group is a group.

Let S be a set and let $F\langle S\rangle = \{\phi : S \to \mathbb{Z}\mid \phi(x) = 0\ \text{ for all but finitely many } x \in S\}$. Show that $F\langle S\rangle$ is an abelian group w.r.t. the ...
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-3 votes
0 answers
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For every abelian group $A$, is $n A / m A$ isomorphic to $A / (m - n) A$?

Let $A$ be an abelian group. For arbitrary positive integer $n$, $m$, I think $nA / mA$ and $A / (m-n)A$ are isomorphic as abelian groups. $$ nx \;\mathrm{mod}\, mA \mapsto x \;\mathrm{mod}(m-n)\, A ...
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Decomposition of $L^2(G)$

Let $G$ be a compact group. By Peter-Weyl, we have the following decomposition $$L^2(G) = \bigoplus_{\pi \in \hat{G}} V_\pi, $$ where $$V_\pi = \text{span} \{ x\mapsto \langle \pi(x)u,v \rangle \in L^...
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1 answer
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If $G$ is a group and $|G|=6$ show that there exists a $a \in G$ such that $O(a)=6$ [closed]

$O(a)=|\langle a\rangle|$, my professor states that $a \in (G-\{e\})$ so the Lagrange theorem implies to $O(a)>1$ and $O(a)|6$, I'd like to understand just this part of the proof which I was given ...
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3 votes
2 answers
50 views

Perfect groups, central extensions and abelian groups

Let $ G $ be a perfect group. We often consider extensions $$ 1 \to A \to E \to G \to 1, $$ where $ A $ is abelian, $ G $ is perfect and the image of $ A $ is central. Is it possible for such an ...
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0 answers
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Step in proof showing $p$-groups are product of cyclic groups

I am trying to follow a proof that shows that an abelian group of order $p^n $ for some $p$ prime and $n\geq 1$ then $G$ is a product of cyclic groups. We have chosen $a \in G$ to have maximal order ...
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3 votes
0 answers
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Generators of lattice

Consider two lattices $\Lambda_1\subseteq\Lambda_2$ in the complex plane, both generated by two elements, respectively. Is it always possible to find generators $\alpha$ and $\beta$ of $\Lambda_2$ ...
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0 votes
1 answer
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Module Homomorphisms and the nature of abelian groups

There is a fundamental result specifying that all abelian groups are $\Bbb Z$-modules. We are trying to use this fact to find out: Does there exist an abelian group $H$ which is not isomorphic to $\...
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1 vote
0 answers
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Automorphism and cyclicity

Let $G$ be a group. Show that if ${\rm Aut}(G)$ (the group of automorphism of $G$ ) is cyclic, then $G$ is abelian, and if $G$ is additionally finite, show that $G$ is cyclic. I need the second part ...
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1 vote
0 answers
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Existence of infinite chain of $p$-th roots of elements implies trivial group

Let $p$ be a prime and let $H$ be a subgroup of an infinite abelian group $\Gamma$, and suppose we have a chain of elements in $H$ $$ h_0, h_1, \ldots $$ such that $h_{n+1}^p = h_n$. Does it follow ...
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1 vote
0 answers
53 views

Divisibility of order of odd primes

I am having hard time showing the above. I thought of using Euler's criterion but I can not proceed from $q-1\nmid w_n/2$ and $p-1| w_n/2$
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3 votes
1 answer
47 views

In sage: from $\mathbb{Z}/N\mathbb{Z}$ to $(\mathbb{Z}/N\mathbb{Z})^*$

In Sage G=IntegerModRing(11^5-1) gives me the cyclic group $\mathbb{Z}/161050\mathbb{Z}$. From here U=G.unit_group() gives me ...
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1 vote
0 answers
76 views

Prove that if $x^3 = e$ then $\mathcal G$ is an Abelian group. [closed]

I am given that for all $x$ in $\mathcal G$ (where $\mathcal G$ is the group) the following equality holds: $$(ax)^3 =x^6$$ for some $a$ in $\mathcal G$. I need to show that $\mathcal G$ is Abelian. I ...
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0 answers
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Proof that special orthogonal group SO(3) is not abelian group [duplicate]

I have to show that $\operatorname{SO}(3)$ is not an abelian group. I tried proving these points: $A^{-1}$ exists $\forall A \in \operatorname{SO}(3),$ if $A,B \in \operatorname{SO}(3)$, then $AB\in \...
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2 votes
1 answer
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Equivalent of floor division in a group of integers mod N.

I'm working on a small programming project, and I'm struggling a bit with calculating fractions of numbers in a commutative group. I'm by no means a mathematician or a programmer, so please bear with ...
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1 vote
1 answer
26 views

Basis of $G\otimes_{\mathbb{Z}}\mathbb{R}$ for a finitely generated abelian group $G$.

Suppose that $G$ is a finitely generated abelian group, i.e. there exists a finite generating set $\{x_{1},...,x_{n}\}$ in $G$ such that every element of $G$ can be written as $\sum_{i=1}^{n}\lambda_{...
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3 votes
1 answer
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Finite abelian group and its automorphism

Let $G$ be a finite abelian group with $x, y \in G$ such that $|x| > 2$, $|y|$ divides $|x|$ and $y \notin \langle x \rangle$. Then there exists no automorphism $f$ on $G$ such that $f(x) = x$, $f(...
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3 votes
1 answer
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Use injectivity of $\Bbb Z/n\Bbb Z$ over $\Bbb Z/n\Bbb Z$ to prove Prüfer's First Theorem

I'm working on the part (b) of the exercise at P.67 of Brown's Cohomology of Groups, which is stated as follows: Let $R = \mathbb{Z}/n\mathbb{Z}$. Show that $R$ is an injective $R$-module. Deduce: (a)...
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3 votes
1 answer
102 views

Let $G$ be a finite, abelian group, prime $p\mid |G|$ and $G$ contain $p-1$ elements order of $p$. Prove $G$ is cyclic.

If $G$ is a finite abelian group, then TFAE: $G$ is cyclic if $p$ divides order of $G$ then $G$ contains exactly $p-1$ elements of order $p$? The forward direction is easy but for the other part I ...
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0 votes
0 answers
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Questions clarification needed for showing that $H$ is an abelian group of odd order.

For the following exercise, I have some question about the notation and hint: Let $(G, *)$ be a finite group of even order. Suppose that half of the elements of $G$ are of order 2 and the rest of the ...
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0 votes
1 answer
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Let $G_1=G/Z(G)$ for a group $G$. Let $G_{i}$ be the group $G_{i-1}/Z(G_{i-1})$. If $G_{k}/Z(G_{k})$ is abelian, is $G$ nilpotent? [closed]

Let $G$ be a group and $Z(G)$ be its center. Now let $G_{1}$ be the group $G/Z(G)$ and $Z(G_{1})$ be its center. Inductively let $G_{i}$ be the group $G_{i-1}/Z(G_{i-1})$. If for some $k$ we have ...
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1 vote
1 answer
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$G$ has an abelian subgroup $K$ with $|K|\geq p^3$.

Let $G$ be a group of order $p^4$ for some prime number $p$. Suppose $G$ has a normal subgroup $H$ of order $p^2$. Then $G$ has an abelian subgroup $K$ with $|K|\geq p^3$. First, by acting $G$ onto $...
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0 votes
2 answers
42 views

Uniquness and existence of free abelian group and construction

Let $I$ be a set. A pair $(G,\epsilon)$ consisting of a abelian group $G$ and a map $\epsilon:I \to G$ is called a free abelian group over $I$ if and only if for all abelian groups $H$ and maps $\phi :...
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2 votes
1 answer
180 views

$\mathbb Q/\mathbb Z$ and $\mathbb R/\mathbb Z$ are injective cogenerators for $\mathbb{Z}\textbf{-Mod}$

I wish to show that The abelian groups $\mathbb Q/\mathbb Z$ and $\mathbb R/\mathbb Z$ are injective cogenerators for the category $\mathbb{Z}\textbf{-Mod}$. Recall that An object $E$ in a category ...
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4 votes
1 answer
84 views

In a group, when does $|ab| = \mathrm{lcm}(|a|, |b|)$

In a finite abelian group $G$ where $a$ has order $m$ and $b$ has order $n$, I was able to prove that $\mathrm{lcm}(m,n) \mid |ab|$ by proving that $(ab)^{\mathrm{lcm}(m,n)} = e$. I know that it is ...
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0 votes
0 answers
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Does every proof that abelian groups are amenable rely on the axiom of choice?

Does every proof that abelian groups are amenable rely on the axiom of choice? So far, any proof I've seen that all, say countable discrete, abelian groups are amenable requires some sort of argument ...
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1 vote
1 answer
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Commutative diagram involving divisible group

I am working on the exercise below from an old commutative algebra qualifying exam. I've been studying Atiyah & Macdonald's commutative algebra text. Let $\pi:\mathbb{Q} \rightarrow \mathbb{Q}/\...
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-1 votes
1 answer
33 views

The cartesian product of an abelian and non abelian group [closed]

If I take the Cartesian product of two groups, with one being abelian and the other being non abelian, Is the product always abelian, always non abelian, or can it be either?
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1 vote
2 answers
48 views

The equational and quasi-equational theory of commutative groups in the signature $*$.

I know that the equational theory of groups in the signature $*$ is axiomatized by the associative law, so does that mean the equational theory of commutative groups in the signature $*$ is ...
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1 vote
0 answers
59 views

Torsion free group is Abelian if $(gh)^r=g^rh^r$ [closed]

Let $G$ be a torsion free group where for some fixed $r\ge2$ we have the identity $(xy)^r=x^ry^r$ for all $x,y\in G$. Does this imply $G$ is Abelian?
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0 answers
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What can we say about $a, b \in G$ when $na = nb$ with $G$ a $p$-group and $(n, p) = 1$?

Let $a, b \in G$, with $G$ an abelian $p$-group. Define $n$ such that $(n,p)=1$ Assume $na=nb$. Then, I think it follows that $a=b$. I think if $G$ is cyclic, $na=nb \Rightarrow na \equiv nb \pmod{p} \...
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3 votes
2 answers
174 views

A conditioned morphism in a group

Let $(G,+)$ be an abelian group of at least $3$ elements and $f:G \to G$ a homomorphism such that $f(x) \in \{0, x, -x\}$ for all $x \in G$. Show that $f \in \{0_G, 1_G, -1_G\}$. I tried proving that $...
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14 votes
2 answers
873 views

Cannot become ring because distribution law does not hold

Commutative ring with unit is defined as $(R,+,\times)$, where $(R,+)$ is abelian group and $(R,\times)$ is commutative multiplicative monoid with $1$ and $+$ and $\times$ satisfies distributive law. ...
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  • 574
1 vote
1 answer
73 views

In group-theory, are the elements in a set other sets or are they precise numbers?

I am starting out with group theory for my computer science degree, it's part of the basic maths subject, it is covered in about 1.5 pages and then moves on with topology. From what I understood you ...
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0 votes
0 answers
25 views

Maximum order of an element in an Abelian group

I needed to find the number of Abelian groups of order $n = 2^3 \cdot 3^4 \cdot 5^6 \cdot 7^{11}.$ And then to find in how many of those groups the maximum order of an element is $\leq$ $n \div 12$. I ...
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-1 votes
2 answers
42 views

Prove finiteness of quotient group [closed]

Let $G$ be a finitely generated abelian group. Prove that the quotient group $G/2G$ is finite. I tried two approaches but did not succeed. Structure Theorem: $G\cong\mathbb{Z}^r\times\prod\mathbb{Z}...
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