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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2answers
28 views

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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1answer
33 views

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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2answers
79 views

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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1answer
24 views

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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0answers
27 views

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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1answer
21 views

Question about a step in the proof of the uniqueness of the decomposition of a finitely generated R-module, R a PID.

The theorem is the following: Let $R$ be a principle ideal domain, and let $M$ be a finitely generated $R$-module. Suppose that $M \cong R^s \bigoplus R/Ra_1 \bigoplus ... \bigoplus R/Ra_u$ (1) $M \...
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1answer
34 views

Conditional bounds on generating fractions

Let's define $n$-th generating fraction of a finite group $G$ as $gf_n(G) = \frac{\{(a_1, ... , a_n) \in G^n| \langle \{a_1, ... , a_n\} \rangle = G\}}{|G|^n}$. The trivial properties of this object ...
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0answers
44 views

Minimal generator of group

Suppose I have a group presentation $G = \langle a,b,c|a^{2}\rangle$ (i.e. $a = a^{-1})$. I am trying to find a minimal generating set of $G$. So far I have $G = \langle\{ab,b^{-1}c,c^{-1}\}\rangle$ ...
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1answer
42 views

$[G,G] \leq \langle g_1,…,g_{n-1}\rangle$ if and only if $\langle g_1,…,g_{n-1}\rangle$ is normal in $G$

Let $G$ be a group generated by: $\{g_1,...,g_n\}$, then $[G,G] \leq \langle g_1,...,g_{n-1}\rangle$ if and only if $\langle g_1,...,g_{n-1}\rangle$ is normal in $G$. I think I have to prove that $...
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2answers
72 views

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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0answers
44 views

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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1answer
48 views

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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1answer
30 views

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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1answer
31 views

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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1answer
30 views

Working out the structure of a finitely generated field

If I want to work out write the structure of the elements in the field $\mathbb{Q}(\omega,2^{1/3})$, where $\omega$ is the 3rd root of unity - what is the easiest way to do so? Since $\omega$ and $2^{...
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2answers
46 views

Subrings of $\mathbb{Q}$ that are not finitely generated.

Exercise. Let $A$ be a subring of $\mathbb{Q}$ such that $\mathbb{Z}\subsetneqq A$. Then $A$ is not finitely generated as $\mathbb{Z}$-module. I was trying in the following way, but unfortunately I ...
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1answer
49 views

Let $G=\langle x,y|x^{8}=y^{2}=e,yxyx^{3}=e \rangle$. Show that $|G|\leq16$. Assuming that $|G|=16$, find the center of $G$ and the order of xy. [closed]

Let $G=\langle x,y|x^{8}=y^{2}=e,yxyx^{3}=e \rangle$. Show that $|G|\leq16$. Assuming that $|G|=16$, find the center of $G$ and the order of xy. I am having trouble with this proof. I have a hint but ...
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1answer
47 views

This type of group of functions is never finitely generated. Proof?

Suppose there is not finitely-generated group $G$ and a finite set $X$. Then it is impossible that the set of all possible maps $f : X \to G$ is finitely generated. Proof. If $H = \{ f : X \to G \}$...
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2answers
31 views

Proving or disproving that a nonidentity cyclic group has at least two generators. [duplicate]

I had trouble proving that a nonidentity cyclic group has at least two generators, but I am starting to think that it has to be disproven. Would I have to disprove it by showing that any cyclic group ...
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0answers
25 views

“Minimal” type rank of finitely generated abelian groups with torsion

I've been having trouble finding resources because searching up "rank" for abelian groups always brings up the free abelian rank, which is not what I'm looking for. However I'm interested in the more ...
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1answer
60 views

Let $A$ be a PID, $M$ an injective finitely generated module. Prove that $M = 0$. [closed]

Help! Let $A$ be a PID, $M$ an injective finitely generated module. Prove that: $$ M = 0.$$
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1answer
58 views

Finitely Generated Modules - Nakayama's Lemma proof

I am reading "Introduction to Commutative Algebra" written by Michael Atiyah; In the Finitely Generated Modules section, there's a corollary that from Proposition 2.4 which I don't understand its ...
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1answer
46 views

Show that the set of polynomials in $\mathbb{Q}[X]$ such that $f(\mathbb{Z}) \subset \mathbb{Z}$ is not Noetherian.

This problem comes from Bosch’s Commutative algebra and Algebraic geometry, Exercise 1.5.10. Let R be the ring of all polynomials $f \in \mathbb{Q}[X]$ such that $f(\mathbb{Z}) \subset \mathbb{Z}$. ...
4
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1answer
60 views

Does there exist a two-generated simple non-abelian group with specific properties?

Does there exist a simple non-abelian 2-generated group $G$ and two elements $a, b \in G$, such that $\langle \{a, b\} \rangle = G$, $a^2 =1$ and $\forall c, d \in G$ $\langle \{c^{-1}bc, d^{-1}bd \} \...
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1answer
27 views

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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0answers
45 views

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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46 views

Are all almost virtually free groups word hyperbolic?

Suppose $G$ is a finitely generated group with a finite symmetric generating set $A$. Lets define Cayley ball $B_A^n := (A \cup \{e\})^n$ as the set of all elements with Cayley length (in respect to $...
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1answer
34 views

If $\{e_1,…,e_n\}$ is a $\Bbb Z$-basis of $\Bbb Z^n$ then $\{e_1\otimes 1,…,e_n\otimes 1\}$ is a $\Bbb Q$-basis of $\Bbb Z^n\otimes \Bbb Q$

Suppose $\{e_1,...,e_n\}$ is a basis of $\Bbb Z^n$ over $\Bbb Z$ (not necessarily the standard basis). It is well-known that $\Bbb Z^n\otimes \Bbb Q$ is an $n$-dimensional $\Bbb Q$-vector space. I ...
3
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1answer
45 views

Torsion in finitely generated modules over polynomial rings

I've been getting a little lost in algebra today. Let $M$ be a finitely generated $R[x]$-module where $R$ is a PID. There is a short exact sequence $$0\to tM \to M \to F \to 0 $$ where $tM$ is the ...
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1answer
103 views

Can a quotient of a polynomial ring by $n-1$ polynomials make $n$ variables $m$th powers?

Let $K$ be a field, consider the polynomial ring in $n$ variables $R = K[X_1, \ldots, X_n]$ and let $F_1, \ldots, F_{n-1} \in R$ be arbitrary given polynomials. Let $S$ be the quotient ring $R/(F_1, \...
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3answers
40 views

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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0answers
36 views

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 '$A_i$...
2
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1answer
27 views

What does $S[\lambda_1,…,\lambda_n]$ look like? Where $S$ is a subring of a given ring $R$ and $\lambda_1,\ldots,\lambda_n\in R$.

Let $R$ be a ring and $S$ be a subring of $R$. Let $\lambda_1,...,\lambda_n \in R$. I encountered $S[\lambda_1,...,\lambda_n]$, which is the smallest subring to contain both $S$ and $\{\lambda_1,...\...
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2answers
43 views

Direct union of finitely generated subgroup

I'm reading Wagon's book "The Banach-Tarski paradox". At page $149$, the author talks about the "direct union of a directed system of amenabile groups" without defining it. Q1: does direct union ...
4
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1answer
39 views

Subgroup generated by commensuration class of an element of a virtually free group

Suppose $G$ is a group. Let’s call $a, b \in G$ commensurate, iff $\exists m, n \in \mathbb{Z} \setminus \{0\}$, such that $a^n = b^m$. One can see, that commensuration is an equivalence relationship: ...
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1answer
34 views

Is the ideal generated by $a$ simply $\{ au\mid u\in R, a\in I \}$ ($R$ is a ring)

More specifically, $R$ is a commutative ring. I'm trying to understand what the ideal "generated by $a$" is, where $a$ is an element of $R$. I believe this ideal is simply the set $\{a\cdot u\mid u\...
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0answers
63 views

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 ...
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0answers
18 views

modules finitely generated over a local commutative ring

Let $R$ be a local commutative ring, that is, $R$ has a unique ideal maximal $J$. Let $M$ a finitely generated $R-$module such that $MJ = M$. Show that $M = 0$. if assume by contradiction that $M\neq ...
1
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1answer
41 views

Generators of $C_3\rtimes C_2$

Can I write elements of $G=C_3\rtimes C_2$ as $$\{(0,0),(0,1),(1,0),(1,1),(2,0),(2,1)\}?$$ Then, what are the generators of $G$? $(0,1)$ and $(1,0)$? I've learned that the multiplication of semi-...
4
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1answer
92 views

Second derived subgroup of Baumslag Solitar group BS(2,3)

I am thinking about the group $G=BS(2,3)=\langle a,b\mid b^{-1}a^2b=a^3\rangle$. Is it true that $G''\cap\langle a\rangle=1$? By $G''$ I am referring to the second derived subgroup of $G$. If it is ...
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1answer
46 views

Generators of alternating group $A_n$ for odd and even $n$

Assume $n\geq3$ and let $A_n$ be the alternating group of $\{1,\ldots,n\}$. I would like to demonstrate the following claims: $A_n=\langle(123),(12...n)\rangle$, if $n$ is odd; $A_n=\langle(123),(23.....
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0answers
27 views

On the definition of finitely generated algebras

Let $R$ be a commutative ring with unity, I have seen that $A$ is a finitely generated $R$-algebra if and only if it is isomorphic to a quotient ring of the form $R[X_1,\dots,X_n]/I$ by an ideal $I\...
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1answer
38 views

Finitely generated over a ring $S$ and a map $R \to S$ implies finitely generated over $R$?

Let $R, S$ be commutative rings. Suppose I have a finitely generated $S$-algebra $A$. Let $\phi: R \to S$ be a ring homomorphism. Then does this turn $A$ into a finitely generated $R$-algebra? Thank ...
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2answers
70 views

Finitely generated group unlikely to be generated by randomly chosen elements.

A well known interesting fact is that if two integers are picked "at random" (in an appropriate asymptotic sense), the chances they generate the integers is $6/\pi^2$. So, the integers can be ...
2
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1answer
38 views

Set of elements that do not generate $G$ with another one.

Consider a finite $2$-generated group $G$, not cyclic. Fix $a\in G$. Let's denote by $J_a$ the set of elements of $G$ that do not generate $G$ together with $a$, i.e. $$J_a=\{g\in G: \langle g,a \...
3
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1answer
90 views

Is there a criterion for which $BS(m,n)$ are solvable (and non-solvable)? If not, are there classes of such groups where this is known?

Let $BS(m,n) = \langle a,t\mid ta^mt^{-1} = a^n \rangle$ be a Baumslag-Solitar group, with $m,n \in \mathbb{Z}.$ Is there a criterion for which $BS(m,n)$ are solvable (and non-solvable)? If not, ...
0
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0answers
35 views

Finding a generator element of a finite extension field

I would like to know, if there are any intuitive fast approaches to finding generator elements of small finite extension fields. Like for example, i don't want to try every element of lets say $\...
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0answers
27 views

Question about viewing a finite dimensional $k$ vector space as a $k[t]$-module and applying classification theorem for fin. gen. modules over PIDs.

From Algebra: Chapter $0$ by Aluffi If $R=k$ a field, and $M=V$ a $k$-vector space, then we have However, I am bit confused. How come we do no have $$V \cong k[t]^{\mathrm{rk} V} \oplus k[...
2
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0answers
67 views

Proof involving generating sets of groups

I need to prove the following: Let $G$ be a group, generated by the elements $g_1, \ldots ,g_n $. In other words, a subgroup $\langle g_1, \ldots, g_n \rangle$ consists of all different possible ...

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