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.
737
questions
1
vote
3
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90
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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 ...
- 348
3
votes
0
answers
48
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 ...
0
votes
0
answers
38
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 ...
- 125
3
votes
2
answers
82
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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 ...
- 2,937
1
vote
1
answer
43
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Existence of one finite presentation implies all other presentations are also finite for modules?
Given a finite presentation $A^m\to A^n\to M\to 0$ of M, does it mean for any other presentation $\text{Ker}(f)\to A^k\xrightarrow[]{f} M\to 0$, the relations $\text{Ker}(f)$ is also finitely ...
- 1,958
1
vote
1
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39
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Is the intersection of finite submodules a finite submodule? [duplicate]
If $M$ is an $A$-module and $N,P$ are finite(ly generated) submodules of $M$, is is true that $N\cap P$ is also finite?
I cannot think of a counterexample right now, but I neither see how one would ...
1
vote
1
answer
52
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Minimum size of a minimal generating set of a finite abelian group
I am trying to prove that the minimum size of a minimal generating set of a finite abelian group $G$, denoted $d(G)$, where $G=C_{d_{1}} \times \dots \times C_{d_{k}}$ for $d_{i} \mid d_{i+1}$, is $k$....
- 11
1
vote
0
answers
21
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Noether Normalization and Lie Algebra of derivations on a commutative algebra $A$
In my Lie algebra course, the professor asked us to use Noether's Normalization Theorem and use it to say something about the Lie algebra of derivations over a commutative, associative algebra. Now, ...
- 5,518
3
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0
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36
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Finitely generated abelian groups $G,H$ with the same finite quotients (up to isomorphism) are isomorphic
Two finitely generated abelian groups $G,H$ have the same finite
quotients (up to isomorphism). Prove they are isomorphic.
According to an earlier result, it seems that I need to use the fact that ...
- 1,555
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0
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20
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Unitary Character Group: Book request
I am looking for books discussing unirary character groups. In particular, I am interested in the case where $G$ is a finitely generated abelian group.
I suppose different authors use different ...
- 523
2
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3
answers
111
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Finitely generated group has automorphism mapping between two elements of the same order?
For a finitely generated group $G$, is it always the case that if two elements $g_1, g_2$ have the same order then there is an automorphism that sends one to the other (i.e. $\phi$ such that $\phi(g_1)...
- 1,555
4
votes
1
answer
73
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For $G$ finitely generated, $H$ finite, prove $[G:\kappa(G,H)]<\infty$ where $\kappa(G,H):=\cap_{\phi \in{\rm Hom}(G,H)} \ker\phi$
Let:
$$\kappa(G,H):=\cap_{\phi \in {\rm Hom}(G,H)} \ker\phi$$ where ${\rm Hom}(G,H)$ is the set of all homomorphisms from $G$ to $H$.
I am trying to prove that if $G$ is a finitely generated group, ...
- 1,555
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votes
0
answers
80
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Are there nonabelian groups $G$ s.t. $G=\langle a_1,\dots,a_{n-1}\mid a_i^2=\epsilon,\epsilon^2=1,a_ia_j=\epsilon a_ja_i \textrm{ for }i\ne j\rangle.$
I study a proof of Hurwitz's theorem on bilinear compositions of sums of squares. The proof relies on the following lemma (which is given as an exercise):
Lemma. Let $G=\langle a_1,\dots,a_{n-1}\mid ...
- 1,251
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votes
1
answer
45
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Dimension and length
I was trying to prove the following:
Let $R$ be a Noetherian ring and let $M$ be a finitely generated $R$-module. Then $M$ has finite length if and only if $\mathrm{dim}(M)=0$.
I was able to prove ...
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3
votes
2
answers
107
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A module over PID which is not direct sum of cyclic mdules [duplicate]
I am trying to solve the following qualifying exam problem: “Give an example of a module over a PID that is not isomorphic to a direct sum of cyclic modules. Justify your example”. (Carnegie Mellon, ...
- 401
2
votes
2
answers
104
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Smallest possible cardinality of the generating set of a group
A very useful invariant associated to any group is the smallest possible cardinality of a generating set that generates that group. This is always guaranteed to exist, no matter what the group, at ...
- 7,082
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88
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Showing $2 Q = Q$
I want to show these two statements:
1- Show that $2 \mathbb Q = \mathbb Q.$
2-Show that $\mathbb Q$ is not a finitely generated $A$ module.
For the first statement I do not know how to show it, so if ...
0
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answers
48
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$M$ projective and finitely generated implies $Hom_R(M,R)$ is projective and finitely generated
I'm trying to prove that if $M$ is a left $R$-module projective and finitely generated, then $Hom_R(M,R)$ is a right $R$-module projective and finitely generated. I've seen comments where they use ...
- 51
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28
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A field that is a finitely generated $k$ algebra is also a finitely generated $k$-module
Let $F$ be a field, let $k$ be a subfield of $F$, then $F$ is a finitely generated $k$-algebra iff it's a finitely generated $k$-module.
The $\impliedby$ part is obviuos, so I'm only interested in the ...
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43
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If localizations at generators of $A$ are f.g $B$-algebras so is A
Let $A$ be a ring, $A=(a_1,\dots,a_n)$ with $\sum_{j=1}^nx_ja_j=1$ for some $x_1,\dots,x_n\in A$ and suppose that $A_{a_k}$ are finitely generated $B$-algebras for some ring $B$. The end goal is to ...
- 242
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votes
1
answer
136
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Does a given subset of an infinite group generate a finite or infinite subgroup?
Consider a fixed infinite group, such as $SL(n, \mathbb{R})$. Let $S$ be a given finite subset of the elements of $SL(n, \mathbb{R})$, further suppose that $S$ is closed under inverses. Is there any ...
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The $\mathbb{Z}^r$ part in fundamental theorem of finitely generated abelian groups [closed]
Let $G$ be a finitely generated abelian group. Then:
$G \cong \mathbb{Z}^r\times Z_{k_1}\times Z_{k_2} \times ... \times Z_{k_s}$
where $ r,k_1,...,k_s \in \mathbb{Z}$ and $ k_i|k_{i+1}$ for $1 \leq i ...
- 301
1
vote
1
answer
31
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Subring is integral over finitely generated subalgebra
Let $R \subseteq S$ be unital rings and let $G \leq \mathrm{Aut}_R(S)$ be a finite group of automorphisms of $S$ as an $R$-algebra.
We define the invariant subring: $$S^G = \{a \in S \mid \forall \...
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answers
42
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Equations on Free Groups
I was trying to read an old paper concerning equations on free groups and immediately came to a puzzling statement which made me wonder whether I am fundamentally misunderstanding something or if I am ...
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2
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34
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Determine the invariant factors of $\mathbb{Z^2}$
I want to show that $(G,.)$ is a finitely generated abelian group and determine its invariant factor, where $G=\mathbb{Z}^2$ and the binary operation is $(x_1,y_1).(x_2,y_2):=(x_1+x_2,y_1+y_2+x_1x_2)$....
- 2,397
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votes
1
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79
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Finitely presented monoid
How to find the elements of finitely presented monoid on Gap with given relations:
[[x^2237, e], [y^2237, e], [z^2237,e],[x^28,y^19z^9],[y^31,x^13z^17],[z^42,x^8y^17],[xy, yx], [xz, zx],[yz, z*y]];
I ...
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answers
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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 subgroup with infinitely many cyclic subgroups of order 2, every one of which is a ...
- 2,565
4
votes
1
answer
140
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Automorphism of $\mathbb{Z}^2\rtimes\mathbb{Z}$
Let $M$ be a matrix in $ \operatorname{GL}(2, \mathbb{Z})$, are there any books/articles that give a description of $\operatorname{Aut}(\mathbb{Z}^2\rtimes_M\mathbb{Z})$?
According to this paper, the ...
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answer
75
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Exact sequence of modules and freeness
Let $R$ be a commutative ring. Let $0 \to M' \to M \to M'' \to 0$ be a short exact sequence of finitely generated $R$-modules.
Show that if $M'$ and $M''$ are free, then $M$ is free.
I assume that we ...
- 121
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votes
0
answers
30
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K is a field,is K[[x]] finitely generated as a K-algebra? [duplicate]
The formal power series just makes me confused.In my intuition,the answer may be no.But the proof is not so clear.This
question is on Kemper's "A Course in Commutative Algebra" Exercise 1.2(...
- 1
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votes
0
answers
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Under what condition on $G$, every descending sequence of retracts of $M$ stops?
Let $G$ be a finitely generated group and $M$ be a finitely generated $\mathbb{Z}G$-module.
My question: Under what condition on $G$, every descending sequence of retracts of $M$ stops?
What I've ...
- 2,565
2
votes
2
answers
66
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Finitely generated abelian groups clarification
I read about finitely generated abelian groups and would be grateful for a clarification.
My lecturer noted these statments:
Each abelian group $G$ such that $a_1,a_2,\cdots,a_n\in A$ are been chosen,...
- 2,155
10
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1
answer
870
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(Why) Is there no analogue to the classification of finitely generated abelian groups for abelian groups?
Every finitely generated abelian group is a direct sum of cyclic groups. Does this hold for all abelian groups in general? If not, what fails?
4
votes
0
answers
36
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Is there an algorithm to check that a subgroup of a CAT$(0)$ group is *not* quasiconvex?
Let $G$ be a finitely generated CAT$(0)$ group and $H$ a subgroup. If $H$ is quasiconvex then it is finitely generated, so we can immediately conclude that any non-finitely generated subgroup of $G$ ...
- 2,342
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A question on glueing modules
$\require{AMScd}$
Consider the following commutative diagram of integral domains:
\begin{CD}
R_0 @>>> R_1\\
@VVV @VVV\\
R_2 @>>> R_3,
\end{CD}
where $R_3 \simeq R_1 \otimes_{R_0} R_2$...
- 111
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votes
1
answer
64
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Groups with the form $\mathbb{Z}\times G$ with finitely many retracts
A group $H$ is called a retract of a group $G$ if there exists homomorphisms $f:H\to G$ and $g:G\to H$ such that $gf=id_H$. When $G$ is abelian, a retract of $G$ is exactly a direct summand.
By The ...
- 2,565
5
votes
1
answer
135
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Does finitely generated groups have finitely many finite retracts?
A group $H$ is called a retract of a group $G$ if there exists homomorphisms $f:H\to G$ and $g:G\to H$ such that $gf=id_H$.
We know that a group $G$ is finite if and only if $G$ has finitely many ...
- 2,565
3
votes
2
answers
121
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If $G=\langle H,K\rangle$, then $G'=[H,K]$
Let $H,K$ be two Abelian subgroups of a finite group $G$ such that $G=\langle H,K\rangle$. Show that $G'=[H,K]$.
My attempt: Ofcourse, $[H,K]\subseteq [G,G]=G'$. Conversely, let $x,y\in G$. It ...
- 1,495
1
vote
2
answers
59
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Is a finite ring map of finite presentation finitely presented?
Let $\phi: A \rightarrow B$ be a ring map (of commutative unitary rings).
Assume that $\phi$ is finite, i. e. $B$ is finitely generated as an $A$-module, and $\phi$ is of finite presentation as in ...
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1
answer
52
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The set of irrational characters is dense in $\text{Hom}(G,\mathbb{R})$ for finitely generated $G$.
While reading the paper ''residualy finite rationaly solvable groups and virtualy fibring'' by Kielak I am working in the following exercise, which is assumed to be true without explanations, but I ...
- 1,604
1
vote
1
answer
71
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If $aH$ generates $G/H$ where $G$ is cyclic, then $a$ generates $G$.
When $G$ is a cyclic group, any subgroup $H$ of it is also cyclic and hence the quotient group $G/H$ is also cyclic. Moreover, if $a\in G\setminus H$ be a generator of $G$ then $\langle aH\rangle$ is ...
- 3,997
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1
answer
75
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Is it true that the center of a finitely generated nilpotent group is finitely generated? [closed]
Let $G$ be a finitely generated nilpotent group.
Let $Z = Z(G)$ be the center of the group $G$.
Is $Z$ finitely generated?
0
votes
0
answers
29
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Image of finitely generated group in an injective group homomorphism
Suppose I have an injective homomorphism $\varphi:F_n\to F_m$ between free groups, and suppose $G = F_n/\langle r_1, \dots, r_s\rangle$ is some finitely generated group with relations $r_i$. Is it ...
- 1,514
0
votes
0
answers
61
views
$Hom_A(\Omega_{A/K},A)$ finitely generated without torsion?
Consider $A$ a domain which is a finitely generated $K$-algebra for $K$ algebraically closed (say generated by $a_1,\dots,a_n$), define $\tau_{A/K}:=\text{Hom}_A(\Omega_{A/K},A)$, I want to show that ...
- 25
1
vote
0
answers
32
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Two sided ideals of $k\left<x, y\right>$
Question: Are the two sided ideals of $k\left<x,y\right>$ (polynomial ring in twonon commuting variables) finitely generated (as two sided ideals) when $k$ is a field?
I know that there are one ...
- 3,381
0
votes
1
answer
71
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Definition of fractional ideal
I have a little problem with the definition of a fractional ideal. The definition I've been given is a set $f\subseteq Q=\text{Frac}(R)$ such that $\exists b\in R\backslash \{0\}$ such that $b.f\...
- 25
0
votes
1
answer
108
views
If $B$ is an essentially finitely generated $A$-algebra, is the module of Kähler differentials $\Omega_{B/A}$ finitely generated?
I'm studying the module of Kähler differentials from Eisenbud's Commutative Algebra, you can easily get a pdf by typing the name on Google. We have proposition 16.3:
If $\pi:S\to T$ is a surjective ...
- 25
1
vote
1
answer
31
views
Basis of $G\otimes_{\mathbb{Z}}\mathbb{R}$ for a finitely generated abelian group $G$.
Suppose that $G$ is a finitely generated abelian group, i.e. there exists a finite generating set $\{x_{1},...,x_{n}\}$ in $G$ such that every element of $G$ can be written as $\sum_{i=1}^{n}\lambda_{...
- 65
1
vote
1
answer
35
views
Generating the algebra $\mathcal{O}[\mathbf{GL}(n,\mathbb{C})]$ (regular functions)
I recently asked a question about the regular functions in $\mathbf{GL}(n,\mathbb{C})$ but now that I have read the appendix I am again confused; here is my past question: Understanding regular ...
- 567
0
votes
1
answer
37
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Order of Finitely Generated Group of Prime Exponent
Consider the finitely generated group $G=\langle g_1,g_2,...,g_n \rangle=\{g_1^{r_1}g_2^{r_2}...g_n^{r_n}\mid r_i\in\mathbb{Z}\}$. The exponent of $G$ is
$$\exp
G=\text{lcm}\left(\left|g_1^{1}g_2^{1}.....
- 145