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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.

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

The structure of finitely generated abelian groups in particular is easily described. Many theorems that are true for finitely generated groups fail for groups in general. It has been proven that if a finite group is generated by a subset S, then each group element may be expressed as a word from the alphabet S of length less than or equal to the order of the group.

837 questions
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Finitely generated module with a submodule that is not finitely generated

Can someone give an example of a ring $R$, a left $R$-module $M$ and a submodule $N$ of $M$ such that $M$ is finitely generated, but $N$ is not finitely generated? I tried a couple of examples of modules I know and got nothing...
Belgi
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4 answers

subgroups of finitely generated groups with a finite index

Let $G$ be a finitely generated group and $H$ a subgroup of $G$. If the index of $H$ in $G$ is finite, show that $H$ is also finitely generated.
Nana
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Can I recover a group by its homomorphisms?

There is finitely generated group $G$ which I don't know. For every finite group $H$ I can think of, I know the number of homomorphisms $G \to H$ up to conjugation. (By this I mean that two homomorphisms $\phi_1$ and $\phi_2$ are being considered…
Turion
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Show that $({\mathbb{Q}},+)$ is not finitely generated using the Fundamental Theorem of Finitely Generated Abelian Groups.

Can anyone please help me out on how to use the fundamental theorem of finitely generated abelian groups to prove that $({\mathbb{Q}},+)$ is not finitely generated?
Faye
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2 answers

the automorphism group of a finitely generated group

Let $G$ a finitely generated group, $\mathrm{Aut}(G)$ is its automorphism group, then it is necessary that $\mathrm{Aut}(G)$ is a finitely generated group? Thanks in advance.
user151938
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Does every finitely generated group have finitely many retracts up to isomorphism?

The infinite dihedral group $D_\infty = \langle a,b \mid a^2 = b^2 = \text{Id}\rangle $ is a finitely generated group with infinitely many cyclic subgroups of order 2, every one of which is a retract. For the group $\mathbb{Z}\oplus\mathbb{Z}$‎,…
M.Ramana
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If the tensor product of two modules is free of finite rank, then the modules are finitely generated and projective

If over a commutative ring $R$ we have that $M\otimes N=R^n$, $n\neq 0$, need we have that $M$ and $N$ are finitely generated projective? We have finite generation, because if $M\otimes N$ is generated by $\sum_i a_{ij} x_i^j\otimes y_i^j$, then…
Pax
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Why any countable subset of $\mathbb{R}→\mathbb{R}$ is generated by a finite set under composition?

Given a sequence of functions $\{g_k\}$, where $g_k: \Bbb R\to\Bbb R$ for all $n\in \mathbb N$. Prove that there exists a finite set of functions $$f_1,f_2,\ldots,f_n$$ such that any function $g_k$ can be expressed as a composition $$f_{k_1}\circ…
Nikolay Chistyakov
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Finitely generated vs presented

I am curious exactly what are the differences between finitely generated and finitely presented? I understand that finitely generated means we have, for an $R$-module $M$ that there exists an epimorphism $$p:R^n\to M$$ and definitionally that…
Zelos Malum
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Does ZFC decide every question about finitely generated groups?

In ZFC, we can easily say when a triple $\mathscr{G}=\left\langle G,\cdot,1 \right\rangle $ is a group. Furthermore, we can say when a group is finitely generated: First define a "canonical" finitely generated group on $n$ generators by taking the…
Ur Ya'ar
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Finitely generated torsion module over a Dedekind domain

Let $M$ be a finitely generated torsion module over a Dedekind domain $R$. Show that there exist nonzero ideals $I_1 \supseteq \cdots \supseteq I_n$ of $R$ such that $M \cong \bigoplus\limits_{i=1}^n R/I_i$. I'm stuck on this problem. Since $M$…
D_S
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Is almost any group generated by two generators?

What is the asymptotic probability that a randomly chosen finite group can be presented with $2$ generators? More precisely, what is $$ \lim _{n \to \infty} \frac{\text{number of 2-generated groups of order} \le n}{ \text{number of groups of order}…
Dominik
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3 answers

A fraction field is not finitely generated over its subdomain

I'm looking for proofs of the following fact. Suppose that $R$ is a domain which is not a field with fraction field $K$. Then $K$ is not finitely generated as $R$-module. I know this fact is true, at least, when $R$ is Noetherian and I guess it is…
mr.bigproblem
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$M_a =\{ f\in C[0,1] |\ f(a)=0 \}$ for $a$ $\in$ $[0,1]$. Is $M_a$ finitely generated in $C[0,1]$?

Let $C[0,1]$ denote the ring of continuous functions on $[0,1]$. Consider the maximal ideal $M_a =\{ f\in C[0,1] | f(a)=0 \}$ for $a$ $\in$ $[0,1]$. Is $M_a$ finitely generated? Note: It may seem that $M_a = \langle x-a\rangle$ which is certainly…
Arpit Kansal
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15
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2 answers

Must $k$-subalgebra of $k[x]$ be finitely generated?

Suppose $k$ is a field, $A$ is a $k$-subalgebra of the polynomial ring $k[x]$. Must $A$ be a finitely generated $k$-algebra? Thanks.
wxu
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