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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35 views

Why abelian groups instead of modules in Algebraic Topology

I am studying Algebraic Topology, homology and cohomology to be concrete. I am reading\working through Hatcher, Rotman, Harper and sometimes I combine them with other books when none of them give a ...
1
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1answer
57 views

Non-abelian group which squares to equal the identity element?

Does there exist a non-abelian group $G=\{e,g_1,g_2,...,g_n\}$ with order $n+1$ s.t. \begin{align} (g_1 \dots g_n)^2 = e \end{align} Also, does this change if we say that every element in $G$ is its ...
2
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1answer
30 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 ...
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2answers
56 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
80 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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1answer
28 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 ...
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2answers
86 views

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

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} ...
6
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2answers
914 views

When is the automorphism group $\text{Aut }G$ cyclic?

Let $G$ be a finite group. Under which conditions on $G$ is the automorphism group $\text{Aut }G$ cyclic? More precisely, does "$G$ is abelian" or "$G$ is cyclic" imply "$\text{Aut }G$ is cyclic"?
2
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1answer
65 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 ...
2
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1answer
34 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 ...
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1answer
211 views

A group whose automorphism group is cyclic

Is there an Abelian group $A$ which is not locally cyclic whose automorphism group is cyclic ?
-1
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1answer
191 views

On groups with none of their quotient groups divisible [closed]

Does there exist a group $G$ that satisfies the following conditions: Any proper subgroup of $G$ is contained in a maximal subgroup. There is some $N\unlhd G$ such that $\frac{G}{N}$ is divisible. ...
0
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1answer
29 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
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0answers
32 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 finite generated case. Let $M$ be an free abelian group, $N$ a subgroup ...
8
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1answer
69 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
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1answer
61 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 ...
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0answers
21 views

Following groups are not pairwise isomorphic in spite of having the same order? [closed]

Can someone help me with the following question? "Prove (Z8, +), (D4, ◦) (the group of symmetries of the square) and the quaternion group (Q, ·): Q = {1, −1, i, −i, j, −j, k, −k} are not pairwise ...
2
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2answers
23 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 ...
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0answers
32 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 ...
3
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1answer
46 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
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1answer
54 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, ...
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2answers
68 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 ...
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1answer
33 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 ...
0
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2answers
39 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
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1answer
30 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 ...
3
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1answer
34 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
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1answer
52 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 ...
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1answer
15 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 $n \in \mathbb{Z}$, ...
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0answers
31 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
22 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 ...
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2answers
40 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 ...
0
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2answers
32 views

Q as an additive abelian group has no minimal generating set

Q as an additive abelian group has no minimal generating set. I have done this question according to the solution given on the stack. First I took a minimal generating set S of Q and an element a in ...
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1answer
42 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 ...
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1answer
22 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?
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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 ...
0
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1answer
40 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 ...
0
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1answer
24 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 ...
1
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1answer
9 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 ...
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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 ...
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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 ...
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2answers
180 views

$Aut(G)$ is cyclic $\implies G$ is abelian

I would appreciate if you could please express your opinion about my proof. I'm not yet very good with automorphisms, so I'm trying to make sure my proofs are OK. Proof: Since $Aut(G)$ is cyclic, ...
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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 ...
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1answer
14 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 $H^{ab}$. ...
1
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1answer
62 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 ...
2
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0answers
69 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 ...
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1answer
50 views

What is the definition of linearly independent subset of an abelian group?

What is the definition of linearly independent subset of an abelian group? I know the concept of this, but i don't know how to define this term precisely. Below is what i tried to formulate: ...
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1answer
27 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 ...
1
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1answer
52 views

If $\gcd(|x|,|y|) = 1$ then $|xy| = \mathrm{lcm}(|x|,|y|)$ in an abelian group.

I am trying to prove this If $\gcd(|x|,|y|) = 1$ then $|xy| = \mathrm{lcm}(|x|,|y|)$ in an abelian group. My idea was we have the following $|x||y| = lcm(|x|,|y|)\times gcd(|x|,|y|)$ since we ...
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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 ...
0
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2answers
61 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 ...