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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Is every unitary ring finitely generated?

I'm puzzled by the following: if $R$ is a unitary ring then $R$ is generated by $1_R$, denoted as $$R = \langle 1_R \rangle. $$ can we conclude that every unitary ring is finitely generated? I know ...
Santa-claus 's user avatar
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1 answer
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Find a relation in GAP

I have a finitely presented group $G$ and a finitely generated subgroup $H<G$. GAP computed that $H$ has finite index in $G$. However, PresentationSubgroupMtc(G,H) cannot compute the presentation ($...
QMath's user avatar
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Existence and examples of height functions

Let $\Gamma$ be a commutative group. Let there be a function $h:\Gamma\to[0,\infty)$ with the following properties. For every real number $M$, the set $\{P\in\Gamma : h(P)\leq M\}$ is finite. For ...
Soumik Mukherjee's user avatar
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Conditions for a general formula for intersections of finitely generated ideals in UFDs

What are some conditions to have a general formula for computing intersections of finitely generated ideals in a UFD? $\def\lcm{\mathrm{lcm}}$ $\def\N{\mathbb{N}}$ I would like such a formula as a ...
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Graded pieces are finitely generated as $R_0$-modules

Let $R=R_0\oplus R_1 \oplus \ldots$ be a Noetherian graded ring and $M$ a graded $R$-module such that $M$ is finitely generated as an $R$-module. I want to show that for each $n$, $M_n$ is finitely ...
kubo's user avatar
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If $G\to\mathbb{Z}$ and $H\to\mathbb{Z}$ are surjective homomorphisms prove that exists $G\times H\to\mathbb{Z}$ with finitely generated kernel.

Let $G$ and $H$ be two finitely generated groups. If $\varphi_G:G\to\mathbb{Z}$ and $\varphi_H:H\to\mathbb{Z}$ are surjective homomorphisms prove that exists $\varphi:G\times H\to\mathbb{Z}$ with ...
Marcos's user avatar
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Subgroup presentation for image of group homomorphism

In general, given a finitely presented group, it's not possible to find a presentation for an arbitrary subgroup without some other condition like finite index. I was wondering though if one was given ...
nl08's user avatar
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Doubling radii of balls in groups of polynomial growth

Let $G$ be a group of polynomial growth, let $S$ be a finite generating set for $G$ and let $B_n$ be the set of elements of $G$ given by words of length $\leq n$ in the generating set $S$. Is there an ...
Saúl RM's user avatar
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An ideal between intersection of two ideals and the product of those ideals

I had a similar train of thought as these two questions, Operations on Ideals in Terms of Generators , LCM generators for the intersection of non-principal ideals in a Noetherian UFD , trying to think ...
PPenguin's user avatar
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Finitely generated, nilpotent, torsion-free group that is also radicable

I am currently working with Mal'cev completions, using the following definition: Let $N$ be group that is Nilpotent Torsion-free Finitely generated Then the Mal'cev completion or radicable hull is ...
noparadise's user avatar
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$o(G) \leq o(a)o(b)$?

Let $G$ be a group such that $G = \langle a, b \rangle$. Mainly I wanted to see if $o(G) = \text{lcm}\left(o(a), o(b)\right)$, but I know $D_6$ is a counterexample. But I can't think of any ...
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Terminology: submodule over subring

I've been reading a few texts and have been confused by some terminology regarding (sub)modules and (sub)rings. All definitions of submodules I've encountered have been in relation to the same ring so ...
mathieu_matheux's user avatar
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Generating set for submodule of free module over a PIR

Let $R$ be a commutative principal ideal ring (not necessarily a domain), $S \cong R^n$ a free $R$-module. We know that any submodule $M$ of $S$ has $m := \operatorname{length}_R(M) \leq n \cdot \...
JBuck's user avatar
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On which rings must a finitely generated module be finitely presented? Is there an 'if and only if' characterization for such rings?

As is well known, if $R$ is a Noetherian ring, then a finitely generated module over $R$ must be finitely presented. However, this is not necessarily true for coherent rings. For example, consider $k$ ...
Liang Chen's user avatar
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1 answer
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Are the following definitions of a finitely generated $k$-algebra equivalent?

In my lecture notes for algebraic geometry, an algebra over a field $k$ is defined as a (unital and commutative) ring, together with a ring homomorphism $\lambda:k\to R$ (such a homomorphism preserves ...
Joe's user avatar
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4 votes
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Is this Monoid Finitely Generated?

Let $G$ be a finitely generated abelian group, $H\leq G$, and $S = \{g_1, \dots, g_n\}$ be some finite generating set for $G$. Let $G’$ be the set of negative-free linear combinations of elements of $...
Teddy Astor's user avatar
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Why is order of finitely generated subgroup bounded above by product of orders of elements?

Let $A=\{a_1,a_2,...,a_k\}$ be a subset of an abelian group $G$. Let $\langle{A}\rangle$ be a subgroup generated by $A$. Suppose that each $a_i\in A$ has finite order $d_i$. Then there are exactly $...
cpt.price's user avatar
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1 answer
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Show that $(\Gamma : 2\Gamma)<\infty$ for abelian $\Gamma$ implies that $P\mapsto \frac12P$ cannot go on forever

I was recently reading this question on the relation between Mordell's Theorem and FLT for the $n=4$ case. Reading more carefully Knapp's book, and more precisely the chapter where he discusses the ...
user265131's user avatar
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Is it possible for a normal subgroup of a finite group have greater number of elements in the minimal generating set?

Let $G$ be a finite group, and $1 \lhd N \lhd G$. With $G = \langle A \rangle$ and $N = \langle B \rangle$ be minimal. Is it possible for $|B|>|A| $? Main motivation behind this question was ...
Leon Kim's user avatar
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Epimorphism of torsion module

Let $M$ be a finitely generated torsion module over a PID $D$. Suppose $\varphi:M\to{N}$ is a module epimorphism and suppose that the invariant factor ideals of $M$ are $(d_1)\supseteq{(d_2)}\supseteq....
zinne98's user avatar
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Step in proof of Structure theorem over PID - Jacobson, Basic Algebra I

I'm trying to do exercise 8.6 in chapter 3 of Jacobson's Basic algebra I, which provides a proof of the structure theorem that does not require the Smith Normal form. Let $x_1,x_2,...,x_n$ be a set of ...
zinne98's user avatar
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3 answers
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On the product of two finitely generated $\Omega$-algebras.

Let $\Omega$ be a signature without predicate symbols and let $\mathcal{A}$ and $\mathcal{B}$ be two finitely generated $\Omega$-algebras. Consider their product in the category of all $\Omega$-...
Alex's user avatar
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Is the inverse of a finitely generated $A$-submodule of $\mathrm{Frac}(A)$ still finitely generated?

Context. I recently realized that the definition of a fractional ideal of an integral domain $A$ I knew was probably the wrong one. To give some context, let me give two different possible definitions ...
GreginGre's user avatar
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There is no surjective homomorphism $\Bbb Z^m\to (\Bbb Z_n)^k$ for $n>1$ if $k>m$

Let $n>1$ be an integer and let $G$ be the product of $k\geq 1$ cyclic groups of order $n$, i.e. $G=\Bbb Z_n\times \cdots \times \Bbb Z_n$. Of course there is a generating set of $G$ of order $k$, ...
blancket's user avatar
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5 votes
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Is $N_1+N_2\cong N_1\oplus N_2$ iff $N_1\cap N_2=(0)$?

Let $A$ be a commutative ring with $1$, $M$ an $A$-module and $N_1,N_2$ two submodules. It is easy to see that $$N_1\cap N_2=(0)\implies N_1+N_2\cong N_1\oplus N_2.$$ I am trying to understand if the ...
Jon's user avatar
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Find some subgroups of the symmetric group $S_n$ having more than $k$ generators

As I know the fact that the symmetric group $S_n$ ($n > 2$ to be safe) can be generated by 2 permutations (e.g. $\{(1 2), (1 ... n)\}$), I wonder whether there are any subgroups of $S_n$ that must ...
Vincent J. Ruan's user avatar
1 vote
1 answer
125 views

The group of all bijective functions on S with composition as its binary operation is finitely-generated iff S is a finite set [duplicate]

Here is the problem, I thought it the I've been thinking for a long time, but I still don't have any ideas. Problem: Let A(S) be the group of all bijective functions on S with composition as its ...
Miicky's user avatar
  • 19
1 vote
1 answer
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Is being finitely generated a local property

Searching on this site and others leads to lots of dicussion about localisation at multiplicatively closed subsets of the form $\{f_i^j\}_{j=1}^\infty$ where $\{f_i\}_{i=1}^n$ generate the whole ring ...
DevVorb's user avatar
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0 answers
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Finitely generated module equal to free module of rank $n$? Not just the quotient of free module.

(For commutative algebra) I have a Ring defined as the polynomial Ring over Field $K$: $R = K[x]$. I need to show that the finitely generated module $M$ is equal to a free module of rank $n$ for ...
mad_scientist's user avatar
1 vote
0 answers
69 views

Not sure how to show a module homomorphism to a finitely generated module is an isomorphism.. [duplicate]

I have the question: Let $M$, $N$ be modules over a ring $R$ with homomorphisms $f, g : N \longrightarrow M$ such that $f$ is surjective and $g$ is injective. Show that: (1) $f$ is an isomorphism if $...
mad_scientist's user avatar
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0 answers
62 views

linear isoperimetric inequality implies hyperbolicity

I am trying to find a nice proof that a finitely presented group satisfying a linear isoperimetric inequality implies it is hyperbolic. I came across these lecture notes, Theorem 3.22, but I am having ...
cede's user avatar
  • 613
3 votes
0 answers
35 views

Linearly Compact Module in $R-Mod$

Definition: A module $M$ is called linearly compact if for a family of cosets $\{x_{i}+M_{i}\}_{\triangle}$, $x_{i}\in M$, $\triangle$ is a directed set, and submodules $M_{i}\subset M$ (with $M/M_{i}$...
YSA's user avatar
  • 133
2 votes
1 answer
67 views

Generators and relations for $\mathfrak{sl}(N, \mathbb{C}[t])$

The Lie algebra $\mathfrak{sl}(N, \mathbb{C}[t])$ can be thought of as matrix valued polynomials. As a vector space $$\mathfrak{sl}(N, \mathbb{C}[t]) = \bigoplus_{k =0}^{\infty}\mathfrak{sl}(N, \...
Rodion  Zaytsev's user avatar
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1 answer
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Help with proof of characterization of finitely generated modules over a PID

I am working through a set of lecture notes on abstract algebra, and I am trying to come up with an alternative proof for one of the theorems. However, I am unsure of how to finish it/if it's possible ...
Jon's user avatar
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0 answers
53 views

A module $M$ is finitely generated if and only if any increasing chain $M_i$ of submodules with union $M$ stabilizes.

I found this statement in wikipedia. A module $M$ is finitely generated if and only if any increasing chain $M_i$ of submodules with union $M$ stabilizes. This is essentially the same as this ...
Expialidocius's user avatar
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1 answer
86 views

Is $\mathbb{C}(x)$ a finitely-generated $\mathbb{C}[x]$-algebra [duplicate]

Is $\mathbb{C}(x)$ a finitely-generated $\mathbb{C}[x]$-algebra? Here we can replace $\mathbb{C}$ by any algebraically closed field $k$.
nkh99's user avatar
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1 answer
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Maximal Essential Extension in Finitely Generated Abelian groups

In GTM4, the author asked the reader to give a procedure for calculating the maximal essential extension of $A$ in $B$, where $B$ is a finitely generated abelian group. I've got completely no idea of ...
Jon Snow's user avatar
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4 votes
0 answers
56 views

Consider the group $\mathbb{Z}_2 \wr \mathbb{Z}$, what is $\mathbb{Z}_2 \wr 2 \mathbb{Z}$?

Consider the (Lamplighter) group $(\bigoplus_{n=-\infty}^{n=\infty}\mathbb{Z}_2) \rtimes_\phi\mathbb{Z}$, where $\phi(1)$ "shifts" every element in $\bigoplus_{-\infty}^{\infty}\mathbb{Z}_2$ ...
ghc1997's user avatar
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0 votes
1 answer
61 views

Partition of minimal generating set [closed]

Let $G$ be a finitely generated abelian group with a minimal generating set $S$. Is it possible to find a partition of $S$ to $\{A,B\}$ such that $A$ generates a free abelian subgroup, and $ B $ ...
David's user avatar
  • 21
1 vote
1 answer
46 views

Finitely generated module over non-unital ring

Let $R$ be a non-unital ring, that is, a ring without a multiplicative identity (or a rng if you prefer). If I want to talk about finitely generated modules over $R$, one factible definition is that a ...
Albert's user avatar
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1 vote
1 answer
85 views

If G is a finite cyclic group of size n is generated by g, is it generated by $g^2$?

In the solution to this problem I've been given by my professor, he says that $g^2$ generates G if n is odd but not if n is even. I am confused about this since I came up with the example that $Z/8$ ...
abiH's user avatar
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2 votes
0 answers
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Is a finitely generated non-abelian group embedded in a semidirect product of a torsion group and a torsion-free group?

Let $G$ be a finitely generated group and $Tor(G)$ the subgroup generated by all torsion elements of $G$. Then $G/Tor(G)$ is torsion-free. In the abelian case we have Every finitely generated abelian ...
Greg's user avatar
  • 482
2 votes
1 answer
81 views

Finitely generated group quasi-isometric to finitely presented group

Let $G$ be finitely presented and let $H$ be finitely generated such that $G$ is quasi-isometric to $H$. Show that $H$ is finitely presented. I am not 100% certain about the definition of quasi-...
DrakenPhilisburg's user avatar
5 votes
1 answer
78 views

Finitely generated group where $|g^n|=o(n)$ for all $g\in G$.

This question arose after I gave a general talk on the sub-additive ergodic theorem, where I gave the (simple) proof that in a finitely generated group, if one has a stationary sequence of random ...
anthonyquas's user avatar
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2 votes
0 answers
40 views

Proof of Fundamental Theorem of Finitely Generated Abelian Groups using Divisibility Ordering of Integers

I'm teaching an Advanced Algebra seminar and presented a proof of the Fundamental Theorem of Abelian Groups that followed the proof presented by Herstein in Topics in Algebra. In that proof you look ...
number.seven's user avatar
2 votes
1 answer
93 views

tetration primitive root $q \mod p$

Consider primitive roots $q \mod p$ where $q$ is a prime and $p$ is an odd prime $> 5$. I am looking for such pairs $q,p$ such that every residue $a_i \mod p$ is of the form $$a_i = q^{(v_i)} \mod ...
mick's user avatar
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1 vote
3 answers
434 views

If $R$ is a Noetherian ring, then $R^n$ is Noetherian [closed]

I'm working with a Noetherian ring $R$. As an $R$-module, $R^n = R \oplus ... \oplus R$. I want to show that $R^n$ is Noetherian in the sense that it obeys the ascending chain condition for its ...
Featherball's user avatar
2 votes
0 answers
64 views

optimization problem over finite groups (or at least finitely generated groups)

Have you ever seen any optimization problem over finite groups (or finitely generated groups)? That is, given a group $G$, we want to maximize or minimize a function $f$ over $G$. An example that ...
MohammadJavad Vaez's user avatar
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0 answers
46 views

If an Ideal I is not finitely generated and then I+(a) is not finitely generated.

Suppose of have a ideal $I\subset R$ that is not finitely generated. Then is it the case that the ideal $I+(a)$ is also not finitely generated. I was thinking to assume the contradiction that it is ...
ben huni's user avatar
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3 votes
2 answers
93 views

What is the order of a group given three generators, $a^2=b^3=c^4=1$ and $cb=ac$

A group $G=\langle a,b,c\mid a^2=b^3=c^4=1, cbc^{-1} = a\rangle $ what is the order of the group $G$ give all such possible values. My attempt: Since $cbc^{-1} =a \Rightarrow cb = ac\;\;(*)$, but then ...
IrbidMath's user avatar
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