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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Does there exist an abelian $2$-group of finite exponent that is not a direct sum of cyclic groups?

Does there exist an abelian $2$-group (an abelian group, all of whose elements have order a power $2$) of finite exponent that is not isomorphic to a direct sum of $2$-cyclic groups? The exponent of ...
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Intersection of a Subgroup with an Abelian Subgroup is Normal in the Product

Question: Let $A$ be an Abelian group with $A \trianglelefteq G$, and let $B \leq G$ be any subgroup. Show that $A \cap B \trianglelefteq AB$. [ref: this is exercise 20 on page 96 of [DF] := Dummit ...
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330 views

Torsion subgroup

Prove that in a finitely generated abelian group $G$ the torsion subgroup is a direct summand (from Scott, Group Theory). Clearly, the torsion subgroup is normal because $G$ is abelian, so we have to ...
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Computing Quotient Groups with Infinite Groups

I've asked a similar question: Computing Quotient Groups But now I want to compute a quotient group involving a direct product in which every direct factor is infinite. For example $\mathbb{Z} \times ...
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Computing Quotient Groups $\mathbb{Z}_4 \times \mathbb{Z}_{10} / \langle (2, 4) \rangle$, $\mathbb{Z} \times \mathbb{Z}_{6}/ \langle (1, 2) \rangle$

Let $G/H = \mathbb{Z}_{4} \times \mathbb{Z}_{10} / \langle (2, 4) \rangle$. I know that $|G/H|$ = 4, so $G/H \simeq \mathbb{Z}_{2} \times \mathbb{Z}_{2}$ or $\mathbb{Z}_{4}$. Since $G/H$ has an ...
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having a subgroup of $\mathbb{Z}^{3}$ and wants to show linear independency with parameters

I am stuck with these hard-star exercises for some time now (they are from a book called "Introduction a L'Algebre et L'Analyse Modernes" de M.Zamansky"), if somebody sees the right way, I will be ...
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If L is a subgroup of $\mathbb{Z}^{3}$, linearly independent, linear equations

This exercise is from a book called "Introduction a L'Algebre et L'Analyse Modernes" de M.Zamansky, I attempted to solve. But I don't know if my solutions are correct (they seem too short to be ...
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347 views

Minimal generation for finite abelian groups

Let $G$ be a finite abelian group. I know of two ways of writing it as a direct sum of cyclic groups: 1) With orders $d_1, d_2, \ldots, d_k$ in such a way that $d_i|d_{i+1}$, 2) With orders that are ...
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112 views

How does an odd order group affect the kernel?

Given that G is an abelian group and $\Psi: G\to G$ is a homomorphism, what can be said about the kernel of $\Psi$ if $G$ has odd order?
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homomorphisms on $\mathbb Z\oplus \mathbb Z$

Is it true that a group homomorphism $f:\mathbb Z\oplus \mathbb Z\to \mathbb Z\oplus \mathbb Z$ that is injective must be an isomorphism? I know that non zero homomorphisms $g:\mathbb Z\to \mathbb Z$ ...
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Let $C$ be the commutator subgroup of $G$. Prove that $G/C$ is abelian

Trying to get my head around the commutator subgroup. This is an excercise from Artin's Algebra: Let $C$ be the commutator subgroup of $G$. Prove that $G/C$ is abelian. Here is what I've done: Let ...
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249 views

Torsion module and its socle

A torsion abelian group has nonzero socle and is an essential extension of it. Let $R$ be a commutative ring. If $M$ is a unital, torsion $R$-module with nonzero socle, is $M$ an essential extension ...
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779 views

Finite abelian groups - direct sum of cyclic subgroup

Let $G$ be a finite abelian $p$-group. It is quite elementary to see that if $g \in G$ is an element of maximal order (and thus its span is a cyclic subgroup of $G$ of maximal order) then $G$ can be ...
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Abelian Groups

Can you check next statements, and they are proofs? Statement 1. Lets $A, A_1, A_2$ - are abelian groups and $$A = A_1\oplus A_2.$$ Then $$A/A_1=A_2.$$ Proof: $$A=\{(a_1, a_2)|a_1\in A_1,~~a_2\in ...
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313 views

Is the torsion subgroup the sum or the direct sum of $p$-primary components?

This is written on Page 93 of Derek J.S. Robinson's A Course in the Theory of Groups: Let $G$ be an arbitrary abelian group, $T$ its torsion-subgroup. For an arbitray prime $p$, denote $G_p$ as ...
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959 views

Finite abelian $p$-group with only one subgroup size $p$ is cyclic

My goal is to prove this: If $G$ is a finite abelian $p$-group with a unique subgroup of size $p$, then $G$ is cyclic. I tried to prove this by induction on $n$, where $|G| = p^n$ but was not ...
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$F$ is a free abelian group on a set $X$ , $H \subseteq F$ is a free abelian group on $Y$, then $|Y| \leq |X|$

I am confused by the proof a proposition: $F$ is a free abelian group on a set $X$ and $H$ is a subgroup of $F$, then $H$ is free abelian on a set $Y$, where $|Y| \leq |X|.$ The proof is: ...
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Direct sum of abelian groups

Usually, for a family $\{G_\alpha\}_{\alpha \in A}$ of abelian groups, one defines $$ \bigoplus_{\alpha \in A} G_\alpha \:= \{ a \in \prod_{\alpha \in A} G_\alpha : \sharp \{ \alpha \in A : ...
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The Hopfian property for groups

Let $G$ be a group, which for my purposes would be abelian. To say that $G$ has the Hopf property is to say that every epimorphism of $G$ is an automorphism. Does anyone happen to recall the context ...
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Simple properties of a direct product

I am working on some homework for modern algebra class. The problem I just finished seems relatively easy, but I have learned to be wary of that feeling when it comes to this material. Below are the ...
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159 views

Cauchy's Theorem (Groups) Question?

I'm afraid at this ungodly hour I've found myself a bit stumped. I'm attempting to answer the following homework question: If $p_1,\dots,p_s$ are distinct primes, show that an abelian group of ...
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$|AB|=mn$ if $\gcd(m,n)=1$

If $A$ and $B$ are finite subgroups, of orders $m$ and $n$, respectively, of the abelian group $G$, prove that $AB$ is a subgroup of order $mn$ if $m$ and $n$ are relatively prime. Lagrange's ...
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Product of Sidon sets

Let $G$ be a compact abelian group with dual $\Gamma$. Let $\Lambda \subset \Gamma$ a Sidon set (see the book of Rudin: Fourier Analysis on Groups for the definition). Consider the set ...
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If $H$ is a normal subgroup of $G$ and if both $H$ and $G/H$ are abelian, is $G$ abelian?

Pretty straightforward: If $H$ is a normal subgroup of $G$ and if both $H$ and $G/H$ are abelian, is $G$ abelian?
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Is $(\mathbb{Z},+)$ a solvable group?

Ok, so abelian groups are solvable. And Thm II.8.5 of Hungerford says A group is solvable iff it has a solvable series. (The group may be finite or infinite.) However, I can't seem to find a ...
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f.g. normal subgroup of finite index in abelian groups

Let $G$ be an abelian group. Suppose $G$ has a (normal) subgroup $N$ which is f.g. torsion-free abelian such that $G/N$ is finite. In light of the fundamental theorem for f.g. abelian groups I ask if ...
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343 views

A criterion for a group to be abelian

I noted a discussion on groups being abelian under a certain restriction on powers of elements, e.g. http://tiny.cc/chs45. Maybe this result (probably not too well-known) concludes it all. Let and ...
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If $G/Z(G)$ is cyclic, then $G$ is abelian

Continuing my work through Dummit & Foote's "Abstract Algebra", 3.1.36 asks the following (which is exactly the same as exercise 5 in this related MSE answer): Prove that if $G/Z(G)$ is ...
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Computing a generating set of the kernel of a module

Given a generating set of a $\mathbb{Z}$-module $M \subseteq {\mathbb{Z}_k}^n$, is there a known algorithm to compute a generating set of $\{u \in {\mathbb{Z}_k}^n \, : \, \forall v \in M \quad v ...
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531 views

Group extensions of cyclic groups

Let $A$ be an infinite cyclic group and $B$ be a cyclic group of order $n$. Suppose $$0 \to A \to G \to B \to 0$$ is a short exact sequence of abelian groups. What could $G$ be? It is clear enough ...
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Normal abelian subgroup of a solvable group [duplicate]

Possible Duplicate: A Nontrivial Subgroup of a Solvable Group How to find a normal abelian subgroup in a solvable group? Could someone help me with this proof? Let $G$ be a solvable group ...
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Whence this generalization of linear (in)dependence?

I recently came across a definition of (in)dependence that is supposed to be a generalization of linear (in)dependence among a set of vectors: An element $x$ is dependent on a set of elements ...
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A condition for a subgroup of a finitely generated free abelian group to have finite index

Let A be a free Abelian group of finite rank and B be a subgroup of A such that $A=B+pA$ for some prime number p, then how to prove $B$ is a subgroup of finite index in A? And if $A=B+pA$ holds for ...
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Calculating Hom(A,B)

I have been studying modules and homological algebra as of late but somehow I have missed how to calculate Hom(A,B) for abelian groups, modules and Hom(A,_)/Hom(_,B) for exact sequences. I have no ...
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Is there a simple way to distinguish between group homomorphisms?

More precisely, I am given a function $f:G\to H$ with the promise that it is a homomorphism. Is there an easy way to determine which homomorphism it is without looking through all of its values? For ...
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282 views

Every abelian group of finite exponent is isomorphic to a direct sum of finite cyclic groups?

Can anyone give me a reference to the aforementioned theorem? W. Hodges uses it for an example in his "Model Theory", but I couldn't find anything on it yet. The group may be (let's say, countably) ...
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In what locally compact abelian groups does $\mathbb{Q}$ embed densely?

I know that there is classification of local fields, but here is a closely related question: Can the additive group of $\mathbb{Q}$ be a proper dense subgroup of a locally compact abelian group, whose ...
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A question about reduced p-groups

I need help with an exercise from Kaplansky's Infinite Abelian Groups (Section 9, Exercise 27). He states the problem as follows: Let $G$ be a reduced primary group which is not of bounded ...
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A question about reduced torsion abelian groups

If a reduced torsion abelian group has no cyclic direct summands of order greater than 2, is it an elementary abelian 2-group? Background: I'm trying to classify the groups whose group rings have a ...
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308 views

Torsion free abelian group of rank 1

I find it hard to understand a part of a proof on torsion free abelian groups of rank 1. Let $A$ and $B$ be torsion free groups of rank 1 and of the same type. Let $a'$ and $b'$ arbitrary non ...
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117 views

order of an element of a group and its coset

I find it hard to understand a part of the proof of the existence of any basic subgroup in every abelian torsion group.I'm going to write you the information I think useful. Let $G$ an abelian ...
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Are cyclic groups always abelian?

If a group $C$ is cyclic, is it also abelian (commutative)? If so, is it possible to give an “easy” explanation of why this is? Thanks in advance!
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Is a pure subgroup of divisible group also divisible?

Is this true that a pure subgroup of divisible group is also divisible?
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139 views

A quotient group and its torsion elements

Let $(x_1,\dots,x_r)$ be a non-zero element of $\mathbb{Z}^r$, and let $h$ be the highest common factor of $x_1, \dots, x_r$. Show that: $$ \mathbb{Z}^r/\langle(x_1,\dots,x_r)\rangle \cong ...
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58 views

Is this claim true about extention of $Z$ by $Z$?

Kindly ask this question: Can we say that $\mathbb{Z}$ × $\mathbb{Z}$ is an extention of $\mathbb{Z}$ by $\mathbb{Z}$?
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853 views

Completing an exact sequence

This seemingly simple question has puzzled me for a while: Determine which abelian groups A fit into a short exact sequence $$0 \to \mathbb{Z}_{p^m} \to A \to \mathbb{Z}_{p^n} \to 0$$ (This is ...
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Prove that if $(ab)^i = a^ib^i \forall a,b\in G$ for three consecutive integers $i$ then G is abelian

I've been working on this problem listed in Herstein's Topics in Algebra (Chapter 2.3, problem 4): If $G$ is a group such that $(ab)^i = a^ib^i$ for three consecutive integers $i$ for all $a, b\in ...
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Questions on free abelian groups

Is the following true? $G$ is a torsion free abelian group of rank $>n$. Let $S$ be be a subgroup of $G$ generated by $s_1,s_2, \cdots ,s_n$. (1) If $m_1s_1+m_2s_2+ ...+m_ns_n$ is linearly ...
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Property of abelianization

This is related to this old MO question which wasn't answered properly, though I don't feel I phrased the question in the best way (or posted it on the right site) Define the abelianization of a ...
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Research Paper and Affine Subspace

I was reading a research paper titled Purity and Reid's Theorem by A.Blass and J.Irwin and i have the problem with the explanation of the proof of the first theorem, that is theorem 1.1. In the proof ...