Questions tagged [finitely-generated]

Question 1

Let $G$ be a virtually polycyclic group with subgroups $H$ and $K$. Let $h(\,.)$ denote the Hirsch length. Does the inequality $$h(G) \geq h(H) + h(K) - h(H \cap K)$$ always hold? (I've found this ...

Question 2

I aim to break down the proof of the Jordan–Hölder theorem in the context of finitely generated $R$-modules. Let $R$ be a commutative ring with $1$, and let $M$ be an $R$-module. In the lecture, we ...

Question 3

Let $R$ be a commutative ring with unity and let $R \operatorname{-Mod}$ be the category of $R$-modules. Question: How can we show that for an $R$-module $M$, the following are equivalent? The ...

Question 4

Let $f, g : X\to Y$ be two homomorphisms of a module $X$ over $R$ into a module $Y$ over $R$ such that $f(s) = g(s)$ for every element $s$ of a subset $S$ of $X.$ Prove that $f(x) = g(x)$ for every ...

Question 5

Consider solving a linear system of the form $M\mathbf{x} = \mathbf{b}$ where the coefficient matrix has entries in $\mathbb{Z}[\sqrt{-6}]$, which is a Dedekind domain but not a principal ideal domain....

Question 6

Say we have a finite group $G$ generated by $\langle g_1,g_2,\ldots ,g_n\rangle$ which is a minimal generating set. What information is required from this set in order to determine the whole group? ...

Question 7

As a comment to this question, I was encouraged to ask the following group-theoretic question separately. Consider the three groups $G$, $G'$ and $G''$, and the homomorphisms $$\phi: G \rightarrow G' \...

Question 8

This is a follow-up to my recent question about $G'=\langle S^{-1} S\rangle$, where $S^{-1}S=\{s^{-1}t \vert s,t\in S\}$. $G'$ is a subgroup of $N = \{x\in G \vert x = s_1^{k_1} s_2^{k_2}\cdots s_r^{...

Question 9

Let $G=1+t\mathbb{F}_q[[t]]$ be the unipotent unit group of the ring $R=\mathbb{F}_q[[t]]$ of formal power series. This question is a continuation of a past question, and for more context see another ...

Question 10

I'm studying finiteness properties of groups and in my research I found the following theorem, which is quite useful: $G$ is of type $F_n$ (i.e. there is a CW complex X with contractible universal ...

Question 11

Let $G$ be a finitely generated abelian group. Then, by the structure theorem, there exists an isomorphism of the form: $$ G \cong \mathbb{Z}^r \oplus \mathbb{Z}_{n_1} \oplus \mathbb{Z}_{n_2} \oplus \...

Question 12

Suppose $G$ is a group such that $G = \langle S \rangle$ for some finite subset $S\subset G$. Then let $G'=\langle S^{-1}S\rangle$, where $S^{-1}S = \{s^{-1} t \vert s,t \in S\}$. Of course, in ...

Question 13

Fix a commutative ring with unity $R$. All $R$-algebras are also assumed to be commutative and unital. Given an $R$-algebra $B$, a sub-$R$-algebra $A \subseteq B$ and finitely many elements $b_1, \...

Question 14

In the group of isometries of the plane, let $r$ be a rotation by $\frac{2\pi}5$ radians and let $s$ be a translation. I'd like to find a finite presentation of the subgroup $G=\langle r,s\rangle$. ...

Question 15

Let $k$ be a field. I have seen people use the term "finitely generated $k$-algebra," which looked fine to me at first until I realized the term may have two different meanings. Let $A$ be ...

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Questions tagged [finitely-generated]

An inclusion-exclusion-like inequality for the Hirsch length

Invariance of the length and the factors (up to isomorphism) in the composition series of a module

How can we show that $- \otimes M$ preserves all limits if and only if $M$ is finitely generated projective?

question about generated modules, is my solution wrong? [closed]

Solving systems of linear equations over finitely generated modules of non-PID rings

Information required to determine a group given a minimal generating set

Seifert-Van Kampen and nontrivial elements in amalgamated products [duplicate]

For a group $G$ generated by $S$, what is the normal closure of the group generated by $S^{-1}S$?

$1+ t\mathbb{F}_q[[t]]$ is finitely generated as topological group

Action of group on connected CW complex implies finitely generated.

Why is the structure theorem for finitely generated abelian groups unique up to isomorphism?

For a group $G$ generated by $S$, what can be said about the group generated by $S^{-1}S$?

Is the intersection of co-finitely-generated subalgebras again co-finitely-generated?

Finding a presentation of the group generated by a translation and a rotation

The term "finitely generated algebra"

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