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
32 views

Regarding those $\mathbb{Z}$-modules whose every finite subset generates a finite submodule.

Let $X$ denote a $\mathbb{Z}$-module (aka an abelian group). Then $X$ may or may not satisfy: $(*)$ for all finite sets $F \subseteq X$, the module generated by $F$ is finite. This properly ...
1
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1answer
33 views

Proving that a normal, abelian subgroup of G is in the center of G if |G/N| and |Aut(N)| are relatively prime.

I was trying to prove that a normal, abelian subgroup of $G$, $N$ is in the center of $G$ given that $|\operatorname{Aut}(N)|$ and $|G/N|$ are relatively prime. The official question: Let $N$ be ...
-1
votes
2answers
99 views

Does $(\mathbb{Z} \times \mathbb{Q})/M$ have any element of infinite order? [closed]

Let $\mathbb{Z} \times \mathbb{Q}$ be the group of ordered pairs $(x, y)$ with $x \in \mathbb{Z}, y \in \mathbb{Q}$ under component-wise addition. Fix $m \in \mathbb{Q}$ and let $M \subset \mathbb{Z} \...
0
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2answers
60 views

Can Lagrange's Theorem for algebraic structure apply here?

For a positive integer $n$ let $Φ(n)$ denote the number of elements $r∈\mathbb Z_n$ such that $\gcd(r,n)=1$. Show $Φ(mn)=Φ(m)Φ(n)$ for all $m, n∈\mathbb N$ such that $\gcd(m,n)=1$. The only thing I ...
2
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1answer
42 views

$G$ is an infinite abelian group such that $G \cong H$ for every non trivial subgroup $H$ of $G$ , then is $G$ cyclic?

If $G$ is an infinite abelian group such that $G \cong H$ for every non trivial subgroup $H$ of $G$ , then is $G$ cyclic , or equivalently asking , then is $[G:H]$ finite for every non trivial ...
2
votes
1answer
73 views

$G$ be an infinite group such that $|Aut (G)|=2$ , then is $G$ cyclic ?

Let $G$ be an infinite group such that $|Aut (G)|=2$ , then is $G$ cyclic ? Since $Aut(G)$ is cyclic here , I know that $G$ is abelian , but this is as far as I can get . Please help . Thanks in ...
8
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1answer
81 views

Automorphisms of infinite abelian groups

It is well-known that the map $Aut$ from the class of groups to itself has fixed points. For $n \neq 2$ or $6$, $Aut(S_n) \cong S_n$, $Aut(D_4) \cong D_4$ and if $G$ is a finite non-abelian simple ...
4
votes
1answer
63 views

Why do the characters of an abelian group form a group?

I was reading through Serre's Linear Representation Theory book and encountered a question to show that the set of all irreducible characters of an abelian group form a group. The proof of closure ...
0
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0answers
34 views

Fundamental theorem of finitely generated abelian groups query

Just had a question about how to apply this theorem. If I am only told that a group is of finite order and is Abelian can we use this theorem? Is there a way to ensure it is finitely generated, or do ...
2
votes
2answers
26 views

Question to the proof of: Let $A$ be a finite abelian group and let $g \in A$. Suppose that $\chi(g)=1$ for every $\chi \in \hat A$. Then $g=1$.

Good day, Currently I am working with the book "A First Course in Harmonic Analysis" by A. Deitmar and I am stuck in the beginning of Chapter 5 on the proof to Lemma 5.1.5. I am repeating the few ...
5
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0answers
72 views

On describing a sort of “well-behaved” subgroups of a free abelian group.

I found this question when I tried to figure out what kind of subgroups of a free abelian group behave just as well as in the finitely generated case. Let $M$ be a free abelian group and $N$ a ...
3
votes
1answer
52 views

Is the following equation equivalent to the 2-cocycle condition?

Given a finite abelian group $G$, I'm looking for functions $\rho:G \times G \to U(1)$ such that 1) $~~~~~\rho(g,e) = 1 = \rho(e,g)$, where $e\in G$ is the unit element and such that 2) $~~~~\...
2
votes
1answer
55 views

Show that $U_{14}\cong U_{18}$?

Because $|U_{14}|=|U_{18}|$ and both of them is cyclic and commutative, so i just need define a function $f : U_{14}\to U_{14}$ that f is homomorphism and bijective. $f(1)=1, f(3)=5, f(5)=7, f(9)=11, ...
-4
votes
1answer
35 views

Let $D(G)$ the conmutator subgroup of $G$. If $H$ is a subgroup of $G$ such that $D(G)\subset H$ show $H$ is normal to $G$ [closed]

Let $D(G)$ the conmutator subgroup of $G$. If $H$ is a subgroup of $G$ such that $D(G)\subset H$ show $H$ is normal to $G$. Please, I appreciate any help, since I have some ideas, but those are ...
1
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2answers
69 views

Abstract Mathematics - Group theory and isomorphism

I have been trying to solve two problems, but I am stuck. Can anybody provide me with some links or theory to solve the following problems? The problems are from a study guide and the test exercises ...
0
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2answers
41 views

when is a finitely generated abelian group finite?

I've been asked to show that a finitely generated abelian group G is finite iff $G/pG = \{0\}$ for some prime number $p$, and to find a group such that that is true for all prime $p$. Not really sure ...
0
votes
1answer
35 views

Abelian group structure and structure of Z[i]

Let $F = \Bbb{Z_p}$. For which prime integers $p$ does the additive group $F^1$ have a structure of $\Bbb{Z}[i]$-module? How about for $F^2$? I'm not really sure on how to approach this question, do ...
0
votes
1answer
40 views

Identify the abelian group that has the given presentation matrix

For the presentation matrices $$ \begin{bmatrix} 0 \\ 5\\ \end{bmatrix} , \begin{bmatrix} 1 & 0 \\ 0 & 1\\ 0 & 0 \end{bmatrix}$$ identify the abelian group they represent. For the ...
3
votes
1answer
36 views

When is a centerless group characteristic in direct product with $\mathbb{Z}^n$?

Consider an abelian group $A$ and a centerless group $B$. We can construct the direct product $A \times B$ of these groups, and $Z(A \times B) = Z(A) \times Z(B) = A \times 1 \cong A$. Now, the center ...
4
votes
1answer
55 views

Endomorphism Ring - Definition

Let $G$ be an Abelian group. We may consider the group $\big(\operatorname{End}(G), +\big)$. Next we may endow $\operatorname{End}(G)$ with the composition of functions to make it a ring. Anyway, it ...
1
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1answer
16 views

Construct a free chain complex K

Let $(A_{n})_{n \in \mathbb{Z}}$ be a set of finitely presented abelian groups. Construct a chain complex $\mathbf{K}$, with each $K_{n}$ a free abelian group, such that for each $n \in \mathbb{Z}$, $...
0
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0answers
32 views

Is there a nice characterization for when the torsion subgroup of a group $G$ is a direct summand?

Pretty much just the title. I'm reading Rotman's Introduction to the Theory of Groups, and he gives an example of an abelian group $G$ such that the torsion subgroup (which he denotes $tG$) is not a ...
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0answers
24 views

Equivalent conditions for a group being divisible

I'm being asked to show the following are equivalent conditions of an abelian group $G$: (i) $G$ is divisible (ii) Every nonzero quotient of $G$ is infinite (iii) $G$ has no maximal subgroup I've ...
2
votes
2answers
41 views

Subgroups of $\mathbb{Z}_p^n$

Is there a nice characterization or construction to list the subgroups of $\mathbb{Z}_p^n$, that is, $\mathbb{Z}_p \times \cdots \times \mathbb{Z}_p$ where $\mathbb{Z}_p$ is the cyclic group of prime ...
1
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2answers
40 views

Q as an additive abelian group has no minimal generating set

$\mathbb Q$ as an additive abelian group has no minimal generating set. I have done this question according to the solution given here. First I took a minimal generating set $S$ of $\mathbb Q$ and ...
0
votes
1answer
47 views

Prove that a group with exponent 3 is abelian.

Let be $G$ a group. Is the following statement true? If every $x\in G, x\neq e=1$ has order at most 3 (i.e. $x^3=1$), then $G$ is abelian. I wanted to prove that $xy=yx\ \forall x,y\in G$. $$xy=x1y ...
0
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1answer
23 views

proving the identity for subgroups.

What is the best way to prove that if a group is a subgroup of some other group? Or more precisely how to prove that they have common identity element?
0
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1answer
26 views

What does it mean for a subgroup $H$ of an abelian group $G$ to be less than or equal to $G$?

I am reading through some linear algebra lecture notes and have come across the following notation: $$K \leq G,$$ where $G$ is an abelian group and $K$ is a subgroup of $G$. What does this notation ...
0
votes
1answer
42 views

Intersection of centralizers is normal?

Let $G$ be an arbitrary group, and suppose that $H=C_G(g_1,\ldots,g_n)$ is also the intersection of all centralizers of finite index in $G$, and furthermore $[G:H]<\infty$. Is it true that $H$ is a ...
1
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1answer
10 views

Lifting a decomposition of abelian $p$-groups.

Let $A$ be a finite abelian $p$-group and $x\in A$ an element of order $p$. Assume that have the following exact sequence : $$1\rightarrow \langle x\rangle \rightarrow A\rightarrow B_1\times B_2\...
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0answers
25 views

Can not see the use of Correspondence Theorem

In Herstein`s proof of Fundamental Theorem of Finite Abelian Groups, I don´t see the use of Correspondence Theorem. It says that exist some subgroup $Q$ of G such that $T=Q/B$. But I`m a bit confused ...
2
votes
1answer
106 views

Direct proof that infinite product of copies of $\mathbb{Z}$ is not projective

It is well-known that the abelian group $$A = \prod_{n=1}^\infty \mathbb{Z}$$ is not free (see, for example this MO question), and that over a PID being free is equivalent to being projective (see ...
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3answers
31 views

How to proceed in the proof of this statement.

I'm reading the proof of "Fundamental Theorem of Finite Abelian Groups" in Herstein Abstract Algebra, and I've found this statement in the proof that I don't see very clear. Let $A$ be a normal ...
0
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1answer
15 views

Commutativity of multiplication of cosets of the commutator subgroup

Take a group $H$ with a non-trivial commutator subgroup, and form the quotient group $H^{ab} = H/H'$. Now, take the cosets of the products of elements $a,b$ and $c,d$: $abH'$ and $cdH'$ in $H^{ab}$. ...
2
votes
1answer
109 views

$G$ contains a normal $p$-subgroup

Let $G$ be a non-abelian finite group with center $Z>1$. I want to show that if $G/Z$ is solvable then $G$ contains a normal $p$-subgroup for some prime $p$ with $p\mid |G:Z|$. $$$$ Since $G/...
2
votes
0answers
74 views

infinitely $p$-divisible elements in $A\otimes \mathbb{Z}_p$

Let $A$ be a (possibly non-finitely generated) torsion-free abelian group. Suppose that $A$ contains no infinitely $p$-divisible elements, then does the same hold for $A\otimes \mathbb Z_p$, where $\...
0
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2answers
62 views

Prove a group G is abelian if it satisfies x^2 = x for every x in G

I originally solved this problem by simply noting that x^2 = x implies x=e, so the only element in the group is the identity...but this is wrong. I am now stuck on this idea though and I have tried ...
1
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4answers
88 views

Show that the group is abelian

Let $M$ be a field and $G$ the multiplicative group of matrices of the form $\begin{pmatrix} 1 & x & y \\ 0 & 1 & z \\ 0 & 0 & 1 \end{pmatrix}$ with $x,y,z\in M$. I have ...
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2answers
38 views

Let $L$ be a subgroup of $\mathbb{Z}^3$ of index $16$. What are the possibilities for $\mathbb{Z}^3 /L$?

Let $L$ be a subgroup of $\mathbb{Z}^3$ of index $16$. What are the possibilities for $\mathbb{Z}^3 /L$? Since $L$ has 16 elements, I think it might be $\mathbb{Z}_{16},\mathbb{Z}_2 \oplus \mathbb{Z}...
0
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1answer
36 views

$G \cong \mathbb{Z}_{p^{n_1}}\times \dots \mathbb{Z}_{p^{n_k}}$. If $p=2$ and $n_1>n_2$, prove that $L(G)\cong \mathbb{Z}_2$.

Let $G \cong \mathbb{Z}_{p^{n_1}}\times \dots \mathbb{Z}_{p^{n_k}}$ be a finite abelian $p$-group, in which $n_1\geq \dots \geq n_k$. Define $$L(G)=\{g\in G \;|\;\alpha(g)=g\; ,\forall \alpha\in Aut(G)...
1
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1answer
16 views

Proof that finite abelian groups are the direct sum of p-parts

A proof of this result was given, however I am having trouble understanding why the part in bold is true. Let $A$ be a non-trivial finite abelian group with $|A|$ having distinct prime divisors $...
0
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1answer
16 views

Are there any infinite (virtually) polycyclic groups with lattice orders that are not linear orders?

I am interested in noetherian group algebras, so I am learning about polycyclic groups. Specifically, I want to generalize some ideas that work well with $k[\mathbb{Z}^n]$ utilizing the lattice ...
1
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1answer
64 views

Generating sets and independent subsets in abelian groups.

Definition. A set $\{x_1,...,x_n\}$ of non zero elements in an Abelian group is independent if, whenever there are integers $m_1,...,m_r$ with $m_1 x_1 + \cdots + m_r x_r = 0$, then each $m_i x_i$ is ...
1
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1answer
62 views

$G$ a non-abelian group of order $p^3$. Show that $Inn(G)$ is abelian.

Let $G$ be a non-abelian group of order $p^3$ for prime $p$. Show that $Inn(G)$ is abelian. The center of $G$, $Z(G)$, is of order $p$ (can be seen in this question). I also know that $G / Z(G)=...
0
votes
1answer
29 views

Subgroups of a direct sum.

Let G be a finite abelian group and let $G = G_1 + G_2$ where the $G_i$ are cyclic. Add it is a $p$-group if you like. How do I prove that it isn't the case that, for some $H_i$, $G = H_1 \bigoplus ...
3
votes
1answer
32 views

rank of an abelian group and its embedment into vectorspace

I am confused about the rank of an abelian group. In this wiki page, https://en.wikipedia.org/wiki/Finitely_generated_abelian_group , we have that $\mathbb{Z}^n \oplus \mathbb{Z}_{q_1} \oplus \cdots ...
2
votes
1answer
36 views

Is there a strategy for expressing finitely generated abelian group as the direct sum of cyclic groups?

I know that every finitely generated abelian group can be expressed as a direct sum of cyclic groups. I wondering how easily we can find the cyclic groups given an abelian group. Specifically, one of ...
0
votes
0answers
73 views

Problem from Marcus' Number Fields

I have been stuck on the 27th problem of the 2nd chapter from Marcus' Number Fields. In it we're given $G$, a free abelian group of rank $n$ and its subgroup $H$, which is again of rank $n$. Now, ...
1
vote
1answer
73 views

Homomorphisms $\frac{\mathbb{Q}}{\mathbb{Z}} \longrightarrow \mathbb{Q}$

Can someone please show me a concrete example of a group homomorphism $$ \frac{\mathbb{Q}}{\mathbb{Z}} \longrightarrow \mathbb{Q}, $$ if it exists? I apparently cannot find any of it (except for the ...
1
vote
1answer
40 views

Defining Presheaves on Categories

I'm learning about sheaves and sheaf cohomology with the eventually goal of using these tools to study Riemann surfaces. My references are Forster's Lectures on Riemann Surfaces and Iverson's ...