# 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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### Equivalence for artinian and noetherian vector spaces

I'm trying to prove the next proposition: For a vector space $V$ over a filed $F$, the next are equivalent: a) $V$ has a finite dimension b) $V$ is a finitely generated module c) $V$ ...
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### Proving an isomorphism between finitely generated non-trivial subgroup

Okay, I feel like I start every question this way, but I have an idea of the concepts and need some help actually putting it into practice. I'm working on this question from Groups and Symmetry by ...
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### Showing groups are not divisible

Let G be an abelian group and use additive notation. Call G a divisible group if given x in G and a positive integer m we can always find an element y of G such that my = x. For example, $Q, R, C,$ ...
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### Finding low-index normal subgroups of finitely presented groups in GAP

I'm a new user of GAP looking to use it to find finite-index, normal subgroups of some finitely presented groups. To provide a concrete example, how would I find all the low-index (say index<200) ...
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### The module $\text{Hom}_C(E,F)$ of two finitely generated projective $C$-modules

Let $C$ be an abelian ring and $E,F$ two finitely generated projective modules. Then $\text{Hom}_C(E,F)$ is a finitely generated projective $C$-module. First of all, since $C$ is abelian, the ...
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### kernel of localization isomorphism

Let $M$ be a finitely presented $A$-module and $S \subset A$ be a multiplicative set. Hence have a surjective map $f:A^{\oplus n} \rightarrow M$ Let $S^{-1}M$ be a free $S^{-1}A$-module. Then we have ...
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### A proof that $\mathbb{Q}^+$ is not finitely generated. Is it correct?

Is $\mathbb{Q}^+$ finitely generated? Justify your answer. This is an abstract algebra exercise I've tried to solve, but I don't know whether my answer (proof) is right or not. Proof So, ...
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### The torsion subgroup of a quotient group of a finitely generated abelian group

Let $G$ be a finitely generated abelian group such that $G/G_t$ has rank $n$ and let $H$ be a subgroup of $G$ such that $H/H_t$ has rank $m$. I want to show that $(G/H)/(G/H)_t$ has rank $n-m$. I ...
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### Classifying finitely generated modules over ring

How to characterize every finitely generated module over $\mathbb{Q}[X]/(X^2+1)^3$? I feel somehow we should use the structure theorem for modules over PID but the ring here is not a domain. So I am ...
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### Is the system of subgroups whose quotient is finitely generated, an inverse system?

In a similar fashion to the construction of the profinite completion, let $G$ be a group and let $$\mathfrak{M}=\{H\trianglelefteq G\;|\;G/H\text{ is finitely generated}\}$$ ordered by reverse ...
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### What is the asymptotics of $gf_2(S_n)$?

Suppose $G$ is a finite group. Let's define $n$-th generating fraction of $G$ as $gf_n(G) := \frac{|\{(a_1, ..., a_n) \in G^n| \langle a_1, ..., a_n \rangle = G\}|}{|G|^n}$. What is the asymptotics ...
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### Are all faithful actions of finite rank free groups ping-pong actions?

Suppose, $G$ is a finitely generated group with a finite set of generators $A$. Suppose $G$ is acting on a set $S$. Let’s call such action a ping-pong action iff $\exists$ a collection of pairwise ...
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### Checking that $\dfrac{\mathbb{Z}[x_1,x_2]}{\langle x_1^2\rangle}$ is a $\mathbb{Z}$-algebra

From NPTEL's Commutative Algebra online course, lecture 17 (link here in the time of the video the example is given), it is given the following example concerning a non-finite $\mathbb{Z}$-algebra: ...
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### Number of maximal subgroups in finitely generated amenable groups

The following statement is known to be true: Any subgroup of a finitely generated group lies in a maximal subgroup Proof: Suppose, $G = \langle \{x_1, … , x_n\} \rangle$ is a counterexample. Then ...
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### How should one interpret $R[A_1,..,A_n]$ where $R$ is a unital ring and $A_i\in M_{n\times n}(R)$

How should one interpret $R[A_1,..,A_n]$ where $R$ is a unital ring and $A_i\in M_{n\times n}(R)$? Intuitively, I considered $R[A_1,..,A_n]$ as almost a set of polynomials where '$X_i$' is each '$A_i$...
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### On the growth rate of groups

Let $G$ be a countable group that is finitely generated and let $S = \{s_1, \dots, s_d\}$ be a generating set. Suppose also that $S$ is closed under inverses. Consider now $\Gamma(G,S) = (V, E)$ the ...