Questions tagged [finitely-generated]

For questions regarding finitely generated groups, modules, and other algebraic structures. A structure is called finitely generated if there exists a finite subset that generates it.

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On 2-generated profinite abelian groups

It's true that a 2-generated profinite abelian group is an epimorphic image of $\hat{\mathbb{Z}}\bigoplus\hat{\mathbb{Z}}$? If yes how can I show this?
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1answer
146 views

Whether a group containing a free group is also a free group

Suppose $G$ is a group generated by two elements $s,t$. Suppose $H$ is a subgroup of $G$ such that $H= \langle s^k,t^k \rangle$ is a free group where $k$ is some integer not equal to $1$. Does it ...
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38 views

Isomorphism of finitely generated modules over a PID

Review question from my first course on Algebraic Structures: Let $M, N, P$ be finitely generated modules over a PID. a) Show that if $M⊕M≅N⊕N$, then $M≅N$. b) Show that if $M⊕P≅N⊕P$, then $M≅N$. ...
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Finitely generated submonoids of commutative cancellative gcd monoids

Let $(M,\cdot,1)$ be a commutative cancellative gcd monoid. That is, $\forall a,b\in M\colon a\cdot b =b\cdot a$ $\forall a,b,c\in M\colon (c\cdot a = c\cdot b)\implies (a=b)$ $\forall a,b\in M\colon ...
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67 views

Every infinite finitely generated group $G$ has an infinite quotient with only finite proper quotients

I am trying to prove this statement: Every finitely generated group $G$ that is infinite has a normal subgroup $K$ such that $G/K$ is infinite and has only finite proper quotients. This is a problem ...
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3answers
58 views

F.G. abelian group so that every non-trivial quotient is cyclic

I need to characterize every finitely generated abelian group G that has the following property: $$\frac{G}{S} \text{ is cyclic for every } \lbrace0\rbrace \lneq S\leq G$$ Given the problems before ...
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1answer
47 views

Non abelian groups with a prime number of one-dimensional representations

I am interested in examples of nonabelian groups that are "finitely generated by elements of finite order" that have a prime number of one dimensional representations. For $p=2$ we have $S_N$...
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1answer
92 views

F.G. abelian group so that every quotient is cyclic

I need to characterize every finitely generated abelian group G that has the following property: $$\frac{G}{S} \text{ is cyclic for every } S\leq G$$ I know I am supposed to use the structure theorem ...
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1answer
63 views

how to prove finitely many? which route is easier?

I want to prove that: If $N$ is finitely generated semi-simple $R-$module, then $N$ is a sum of finitely many simple submodules. I know that if $N$ is a finitely generated $R-$module, then that the ...
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53 views

Using finitely generated in proof (2).

I was reading the proof of $(c) \implies (a)$ i.e., (Given any submodule $M \subset N,$ there exists a submodule $M' \subset N$ such that $N = M \oplus M'$) implies ($N$ is a sum of simple modules) ...
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1answer
43 views

How to find generator for intersection of two large subgroups (permutations in $S_{13}$)?

I have $$\sigma = ( 1, 2, 3, 4, 5 )( 6, 10 )( 7, 11 )( 8, 12 )( 9, 13 ),$$ $$\tau = ( 2, 5 )( 3, 4 )( 6, 7, 8, 9, 10, 11, 12, 13 )$$ as my two permutations that generate $G$. How would I find the ...
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1answer
39 views

How to complete this alternative proof for the FTFGAG

I'm trying to prove the fundamental theorem of finitely generated abelian groups. But I cannot an argument involving free abelian group; that is, avoid using the following proposition: (Don't-Use-...
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34 views

Prime Ideal of Power Series Ring is Finitely Generated - Ideal of Constant Terms

Problem: Let $A$ be a commutative ring with identity and $P$ be a prime ideal of the power series ring $A[[x]]$. Let $Q \subset A$ be the set of elements of the form $f(0)$, as $f(x)$ ranges over $P$...
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1answer
112 views

Extending an element in generating set for free abelian group

Let $G$ be a finitely generated free abelian group and $S$ be a minimal generating set i.e. ${\rm rank}(G)=|S|$. If $w$ is a word in $G$ then when it can be extended in a new minimal generating set ...
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1answer
55 views

Subgroups of finitely generated group are not necessarily finitely generated (proof).

In my abstract algebra course, I learned about finitely generated groups. One of the exercises proves that a subgroup of a finitely generated group is not necessarily finitely generated itself. The ...
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1answer
51 views

Showing that the restricted wreath product $\Bbb Z\wr\Bbb Z$ is finitely generated.

This is the first part of Exercise 1.6.15 of Robinson's "A Course in the Theory of Groups (Second Edition)". According to Approach0 and this Google search, it is new to MSE. The Details: ...
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1answer
38 views

generators of affine transformation

I have to prove that the set $A$ of affine transformations $T(u)=au+b$ where $a, b \in \mathbb{F}$ and $a \ne 0$ forms a group under function composition and then I need to find a generating set for $...
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2answers
31 views

Infinite subgroups of $\text{PGL}(2,\mathbb C)$ where each element has finite order

I am interested in finite subgroups of $\text{PGL}(2,\mathbb C)\cong\text{Aut}\,\mathbb C(x)$. Let $G\subset\text{PGL}(2,\mathbb C)$ be finitely generated such that each $g\in G$ has finite order. Is ...
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1answer
45 views

Finitely generated $R$-module as a sum of cyclic submodules

I know that $M$ is finitely generated if there exist $a_1, ..., a_n$ in $M$ such that for any $x$ in $M$, there exist $r_1, ..., r_n$ in $R$ such that $x = r_1a_1 + \dots + r_na_n.$ But if I want to ...
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41 views

Different Definition of Fundamental Theorem of Finitely Generated Abelian Groups

I was reading about abelian groups but confused in Fundamental Theorem of Finitely Generated Abelian Groups, so I searched some sources and become even more confused due to the different types of ...
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1answer
101 views

A group such that all its subgroups are finitely generated

I have been thinking about the following question: Let $G$ be an infinite group with the property that every proper subgroup of $G$ is finitely generated. Can we say that $G$ is always finitely ...
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1answer
19 views

Basic Property of Finite Presentation of Modules

I'm struggling to prove some rudimentary facts about finite presentations of a module $A$ over a ring $\Lambda$, which is defined in Hilton/Stammbach's A Course in Homological Algebra as a short exact ...
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1answer
134 views

using finitely generated in proof.

I am trying to show that:Let $R$ be commutative with unity. if $N$ be a finitely generated $R-$module, then the following conditions are equivalent: (a) $N$ is a sum of simple modules. (b) $N$ is a ...
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1answer
34 views

Is the intersection of a descending chain of finitely generated groups finitely generated

Given a strictly descending chain of finitely generated groups $$ H_1 > H_2 > H_3 > \cdots,$$ is it always the case that $H = \cap_{i=1}^{\infty} H_i$ is finitely generated? I think the ...
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1answer
89 views

Finitely generated group with subexponential growth and surjection onto $\mathbb{Z}$ has finitely generated kernel.

I am trying to solve exercise 6.E.20 from the book of Clara Löh on Geometric Group Theory. Let $G$ be a finitely generated group with subexponential growth that admits a surjective homomorphism $\pi:...
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3answers
44 views

Classifying $ \frac{\mathbb{Z} \times\mathbb{Z}}{\langle(n,n)\rangle}$ via fundamental theorem of finitely generated abelian groups

I am trying to classify, $\frac{\mathbb{Z} \times \mathbb{Z}}{\langle(n,n)\rangle}$, $n=1,2,3,\cdots, $ using fundamental theorem of finitely generated abelian groups. Inspire by @user134824 in ...
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36 views

Determine the generator for the cyclic group formed by the solutions of $x^9 = 1$.

Find the solutions to $x^9 = 1$ and determine the generator for the cyclic group formed by the solutions. The equation can be factored as $(x^3 - 1)(x^6 + x^3 + 1) = 0$ and the solutions are $$\begin{...
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30 views

Minimal presentation length of a universal finitely-presented group?

It is a rather well known fact, that there exist universal finitely presented groups (finitely presented groups, that contain all other finitely presented groups as subgroups). It is a rather direct ...
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relations of tensor product are relations of original rings?

Given two (finitely-generated, not necessarily commutative) rings $A$ and $B$, given by generating sets $R$ and $S$, respectively, and relation sets $R^*$ and $S^*$, respectively. Is the tensor ...
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1answer
35 views

Sylow subgroup of a group generated by two subgroups

Let $G$ be a finite group and $H=\langle A ,B\rangle$ be a subgroup of $G$ generated by subgroups $A$ and $B$ of $G$. Is it true that $H_r=\langle A_r,B_r\rangle$, where $H_r$, $A_r$ and $B_r$ are ...
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40 views

Meaning of finite generation *over a subgroup*

Let $G$ be a group and $H$ a subgroup of $G$. Given a subgroup $A \leq G$, what does it mean for $A$ to be finitely generated over the subgroup $H$? Similarly, what does it mean for $A$ to be finitely ...
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46 views

Finitely generated module and finitely generated algebra over a Noetherian ring

I am given that $R$ is a Noetherian ring and $S \subseteq R[x]$ is a ring. How can I prove that if $R[x]$ is a finitely generated $S$-module then $S$ is a finitely generated $R$-algebra. I am thinking ...
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1answer
62 views

Does every finite group $G$ have a set of generators such that the sum of the orders of the generators is less than or equal to $|G|$?

Does every finite group $G$ have a set of generators such that the sum of the orders of the generators is less than or equal to $|G|$? This is surely true but I am failing to see why. This is easily ...
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2answers
48 views

Understanding how we can make $M$ into an $R[X]$-module by setting $Xm = f(m)$ for $m \in M.$

Here is the question I want to answer: Let $M$ be a finite $R$-module. Show that if $f \in \operatorname {End_R(M)}$ is surjective then it is also injective. Hint: Let $R[X] \cong R^{[1]}$ and make $M$...
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1answer
45 views

Understanding how the hint will prove injectivity.

Here is the question I want to solve: Let $M$ be a finitely generated $R$-module. Show that if $f \in \mathrm{End}_R(M)$ is surjective then it is also injective. And here is the hint I got for the ...
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43 views

Generators of certain $R \subseteq \mathbb{C}[x,y]$

Let $h \in \mathbb{C}[x]$ with $\deg(h)=d \geq 1$ and let $R_{h,y}:=\mathbb{C}+\langle h,y \rangle \subseteq \mathbb{C}[x,y]$. It is known that subalgebras of $\mathbb{C}[x]$ are finitely generated, ...
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51 views

Prove statement about commutative unital ring

I'm trying to solve this problem from my abstract algebra course: Given $A$ a commutative unital ring. Prove that the following two properties are equivalent: Every ideal of $A$ is finitely ...
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44 views

Symmetric group has a minimal set of generators of any size for $n \geq 4$

For $n \geq 4$, I want to show that $S_n$ has a minimal set of generator of size $k$ for $k \in \{2, \cdots, n - 1 \}$, and also that it does not have a minimal set of generator of size $n$. So I ...
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1answer
50 views

How can you prove Nakayama's lemma over nonunital rings using the unitization?

For a nonunital commutative ring $A$, an $A$-module $M$ is called finitely generated over $A$ if there is a finite set of elements $x_1,...,x_n\in M$ such that every element of $M$ is of the form $...
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30 views

Finitely generated modules over a polynomial ring

Let $M$ be a finitely generated module over $\mathbb{Q}[x_{1},\dots,x_{n}]$ but suppose that $M$ is not a finitely generated $\mathbb{Q}$-module. I would like to know if $M$ has a free direct summand ...
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1answer
15 views

Finitely generated module as an (in)finite sum of submodules

Let $R$ be a ring, $M$ be a nontrivial proper left $R$-module and $(M_\lambda)_{\lambda \in \Lambda}$ be a family of submodules of $M$ such that $M = \sum_{\lambda \in \Lambda} M_\lambda$. If $M$ is ...
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28 views

A free $A$-module $M$ which is also finitely-generated. [duplicate]

I have come upon with the following problem: Let $M$ be an $A$-module (assume that $A$ is commutative with identity) which is both free and finitely-generated. My question is whether there is an ...
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1answer
39 views

What is a finitely generated G-operator group?

[I'm self-studying group theory after many, many years away, using Derek Robinson's book.] Robinson in Ex 2.2.8 says: Let G be a finitely presented group and let N be a normal subgroup which is ...
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1answer
79 views

Elementary proof that $\mathbb{Q}$ is not a finitely presented $\mathbb{Q}[x_1,x_2,\dots]$-module

I am trying to prove in an elementary way that $\mathbb{Q}$ is not a finitely presented $\mathbb{Q}[x_1,x_2,\dots]$-module. According to this post, the general case for any non noetherian ring can be ...
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1answer
115 views

Ring is Noetherian if it admits a faithful finitely generated module with ACC on submodules generate by ideals

I have a (commutative unitary) ring $A$ and a faithful finitely generated module $M$ over $A$ which satisfies ACC on submodules of the form $IM$, where $I\subset A$ is an ideal. I want to prove that $...
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1answer
58 views

Quotient group of finitely generated group is finitely generated

I am studying elementary level of algebra, and I'm trying to prove that a quotient group of finitely generated group is finitely generated. It is intuitively true, but I can't get an idea to prove it.
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1answer
95 views

Let $G$ be f.g. with $H\le G$ s.t. $[G:H]<\infty$. Then $\exists K\le H$ with $K\sqsubseteq G$ and $[G:K]<\infty$.

This is Exercise 4.29 of Roman's "Fundamentals of Group Theory: An Advanced Approach". According to Approach0, it is new to MSE. The Details: Definition: A subgroup $H\le G$ is ...
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1answer
85 views

If $C\supseteq A$ is finite over $A$, then any subring $A\subseteq B\subseteq C$ is finite over $A$

Let $K$ be a field and let $A, B,C$ be $K$-algebras such that: $A \subset B \subset C$ $A\subset C$ is a finite extension. $B \subset C$ is a integral extension. So $C$ is a finitely generated $A$-...
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48 views

Not finitely generated homology covered by finitely generated homology

I want to Find a space X which is covered by finitely many open sets $U_i$ such that $H_n(U_i)$ is finitely generated for all n but $H_n(X)$ is not finitely generated. I know how to produce Moore ...
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42 views

Proves about the group $[H,K]=\langle [h,k] : h\in H, k\in K\rangle$

I want to check if my solution to this problem from my group theory course is correct: Let $H,K\leq G$. We define $$[H,K]=\langle [h,k] : h\in H, k\in K\rangle.$$ Prove that $[H,K]=[K,H]$, and that $...

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