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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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1answer
13 views

A question regarding “equality” of word lengths for two minimal generating sets of a finite group

Let $G$ be a finite group $d(G) = \min_{<S>=G}|S|$. Suppose that $|X|=|Y|=d(G)$ and $<X>=<Y>=G$. Let $|g|_X$ be the word length of $g$ with respect to $X$ and $|g|_Y$ be the word ...
4
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1answer
51 views

Subgroup of finitely-generated subgroup

Is there a standard name for this concept: Let $H \leq G$ be groups. Say $H$ is ?? if there is a finitely-generated group $K \leq G$ such that $H \leq K$. What should one use in place of "??"? I'm ...
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1answer
43 views

Is a finitely generated module over the field of fractions is also finitely generated over the original integral domain?

Let $R$ be an integral domain and $F$ its field of fractions. Let $M$ be a finitely generated $F$-module. Question: Is $M$ also a finitely generated $R$-module? I know that $M$ is an $R$-module ...
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1answer
20 views

Systematic approach to find base of vectorspace given its elements' traits

I'm trying to find a base for a vector space that's given as a set with certain traits. Take this example: Let $V$ be an $\mathbb{R}$-vector space with $$ V := \left\{ (a, b, c, d) \in \mathbb{R}^4 : ...
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0answers
14 views

Tensor product of modules finitely generated

Let $S$ be extension of $R$ and $M$ is an $R$-module. If $M$ is finitely generated of $R$-module then $S\otimes_RM$ is finitely generated of $S$-module. Proof We can construct a ring homomorphism $f:...
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27 views

Derived subgroup of a finitely generated nilpotent group [duplicate]

I am looking for a "standalone" proof of the fact that the derived subgroup of a finitely generated nilpotent group is itself finitely generated. I would appreciate either a proof or reference to ...
0
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1answer
18 views

Condition for a finitely generated flat module be projective

Prove that: Let $R$ be a commuatative ring, let $T$ be total quotient ring of $R$. A finitely generated flat $R$-module $M$ is projective if and only if the scalar extension $T\otimes_R M$ is a ...
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19 views

Determining the Cyclic Decomposition of a Finitely Generated Abelian Group given a set of relations.

I am given a group $G=\text{Span}(w,x,y,z)$ with relations defined by $$\begin{bmatrix}0&0&1&3\\-2&1&1&3\\-2&4&1&3\\0&-3&1&5\end{bmatrix}\begin{bmatrix}...
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0answers
11 views

correspondence covering spaces of a free group and its finite index subgroups

Let $F$ be a free group. Let's take a finite covering space $\Gamma$ of the graph $\Delta$ representing $F$ (via its fundamental group so $\pi_1(\Delta) \simeq F$ ). In the proof of Marshall Halls's ...
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1answer
75 views

Necessary and sufficient conditions that $\langle \zeta, (ij), \lvert\lvert k\, \ell \rvert\rvert, \xi_M\rangle$ generates $\mathscr{P}_n.$

Throughout I use cycle notation and write maps $m:X\to Y$ on the right of their arguments (e.g. $xm=y$ for $m(x)=y$). Let $\zeta=(12\dots n)$. This question is inspired by the following questions: ...
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1answer
35 views

Index of the projection of a subgroup on a quotient by finite normal subgroup.

Given a finitely generated group $G$ and a finite normal subgroup $N \leq G$. I am trying to compare finite index subgroups in $G$ and $G/N$. I know that $H$ is a subgroup of $G$ iff $H/N$ is a ...
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0answers
21 views

Uncorrelatedness for random elements of finitely generated groups?

Suppose $G$ is a finitely generated group, $A$ is its finite set of generators. Lets denote the metric induced by the Cayley graph $Cay(G, A)$ on $G$ as $d$. Suppose $\{X_i\}_{n = 0}^\infty$ is a ...
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0answers
26 views

Finitely generated nilpotent group is isomorphic to a quotient of the free nilpotent group.

Let $F^{(r)}$ be the free group generated by $r$ elements. Let $\gamma_n(F^{(r)})$ denote its lower central series. Finally, let $F_{n,r} = F^{(r)}/\gamma_{n+1}(F^{(r)})$ be the free nilpotent group ...
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0answers
26 views

Growth rate of finitely generated nilpotent groups

Let $N$ be a group and $S$ a finite, symmetric generating set with the identity. For $n \in \mathbb N$, we let $S^n = \{s_1\dots s_n\mid s_i \in S\}$ We say $N$ has polynomial growth rate if $\...
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0answers
17 views

Subgroup of ${\rm Aut}\,(\widehat{\mathbb{Z}})$

Ribes and Zalesskii Corollary 4.4.8 show that the group of continuous automorphisms of $\widehat{\mathbb{Z}}$ satisfies ${\rm Aut}\,(\widehat{\mathbb{Z}})\cong\mathbb{Z}_2\times\frac{\mathbb{Z}}{2\...
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0answers
20 views

Finding generators of a group from its action on a topological space

Summary I believe I've written a geometric group theory flavoured proof with a mistake in it, but I'm struggling to see why it might be wrong. I haven't found a counter example, but it also feels too ...
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3answers
33 views

Finitely generated, non abelian, infinite group

I was making a diagram of different types of groups; finite /infinite, cyclic / non-cyclic, finitely generated / inifinitely generated, but realized that I didn't have any examples og infinite groups, ...
3
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1answer
81 views

Prove or disprove: $S_{10} = \langle (1,3),(1,2, … ,10) \rangle$

Prove or disprove: $S_{10} = \langle (1,3),(1,2, ... ,10) \rangle$ I know that $S_{10}=\langle (1,2) , (1,2,...,10) \rangle$. I tried to use this fact to prove the above but failed. It made me think ...
2
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1answer
32 views

Finding generator of a Schnorr group

A Schnorr group is a large prime-order subgroup of $\Bbb Z^*_p$, the multiplicative group of integers modulo $p$. To generate such a group, we find $p = qr + 1$ such that $p$ and $q$ are prime. ...
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0answers
21 views

When are indecomposable projective modules finitely generated?

What conditions to you need to put on your ring to guarantee that the indecomposable projective modules are all finitely generated? Edit: I was hoping there was some general result for this. If your ...
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1answer
34 views

Show the Dihedral Group $D_n$ is generated by rotations and reflection along the x axis.

I'm having problems understanding the excersice: E) Define $D_n$ as the group of symmetries of a regular n-gon. Name the vertices $V=\{V_0,V_1,...,V_{n-1}\}$ so that $$V_{k}=\exp({i\cdot\dfrac{2\pi k}{...
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0answers
7 views

ranks $dim(C_{d}(K), dim(B_{d-1}(K)),dim(Z_{d}(K))$ for K a simplex of n vertices

given n vertices (0-dimensional simplices) consider the full simplex K generated by these n vertices. choose a dimension $d \leq n-1$ I need to prove the closed form formula for the following ranks ...
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0answers
39 views

Generating words in a finitely presented group in SAGE

I'm trying to get a list of all words of length $n$ (in the word metric sense) in some finitely presented group. I have tried some very naive enumerations but it is very slow. Is there an efficient ...
2
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2answers
45 views

if $M \otimes_A (A_m/mA_m)=0$ for every maximal ideal $m \subset A$, then $M=0$, $M$ finitely generated

Suppose $M$ is a finitely generated $A$-module. Prove that if $M \otimes_A (A_m/mA_m)=0$ for every maximal ideal $m \subset A$, then $M=0$. Subscrpit $_m$ means localization at $m$. First consider ...
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2answers
22 views

Need help understanding a step in a proof about modules over PIDs

This is Chapter 3, Theorem 7.3 in Algebra by W. Adkins & S. Weintraub (GTM). It's about the uniqueness of the cyclic decomposition for finitely generated modules over a PID. I highlighted the ...
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1answer
46 views

Why all nilpotent and finitely generated groups are Max?

An exercise asks to prove that if a group $G$ is nilpotent and finitely generated then it satisfy Max condition, in other words all non empty and totally ordered, with respect of inclusion $\subseteq$,...
4
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1answer
51 views

Proving $G \ast_A$ finitely presented $\Leftrightarrow$ $A$ finitely generated

I want to prove the HNN extension $G \ast_A$ is finitely presented $\Leftrightarrow$ $A$ is finitely generated, given that $G$ is a finitely presented group, say $G = \langle S \mid R \rangle$ The $\...
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1answer
29 views

Finite generation of cones in $\mathbf{Z}^n$

Let $X$ be a finite free $\mathbf{Z}$-module of rank $n$ and let $X_+ \subset X$ be a cone i.e. a subset of $X$ such that for any $x,y \in X_+$ and $a,b \in \mathbf{N}$ we have $ax+by \in X_+$. Is ...
2
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1answer
26 views

Isomorphism between finitely generated projective modules over algebra of functions

Here is the context: Consider two finitely generated projective modules $M$ and $N$ over the commutative algebra of (smooth or) continous functions on a manifold $\mathcal{M}$ (they are sections of ...
2
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0answers
43 views

Does a finitely generated group $G$, ever act on $G/N$ freely?

Say that $G$ is a finitely generated group on $k \geq 2$ generators and $N$ is a normal subgroup of $G$. I want to know if I can construct a $G$-action on $G/N$ such that the action is free. None of ...
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0answers
34 views

Set of all R-homomorphism is finitely generated

I have the following proposition: Proposition: If $R$ is a principal ideal domain (PID) and $M$, $N$ two finitely generated (fg) $R-$modules, then $\text{Hom}_R(M,N)$ is finitely generated. My idea: ...
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2answers
35 views

Can I conclude that my group is finitely generated, if it is a homomorphic image of a free-group on finitely many generators?

Say $X$ is a finite set, $F \langle X \rangle$ is the free group on the set $X$ and $G$ be a group. If I have a surjective homomorphism $$\varphi : F \langle X \rangle\longrightarrow G$$ then can I ...
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0answers
34 views

Quick proof that $SL_2(\mathbb Z/n\mathbb Z)\cong \oplus_{p\mid n}SL_2(\mathbb Z/p^{e_p}\mathbb Z)$

I'm looking for a quick proof that $SL_2(\mathbb Z/n\mathbb Z)\cong \oplus_{p\mid n}SL_2(\mathbb Z/p^{e_p}\mathbb Z)$ for some nonnegative integers $e_p$. I've argued as follows, but I'm hoping for a '...
4
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1answer
48 views

Could $\langle \Gamma | R \rangle \cong \langle \Gamma | S\rangle$ if $\langle R\rangle \subsetneq \langle S\rangle$?

If we have two finitely presented groups $\langle \Gamma | R\rangle$ and $\langle \Gamma | S\rangle$ with $\langle R\rangle \subsetneq \langle S\rangle$, could they be isomorphic?
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1answer
13 views

Dimension of span of a finitely generated subgroup

If $QA\subset V$ is the span of $A\subset V$, V- a vector space over the field $Q$, $A$ - a finitely generated abelian subgroup (and hence has a finite rank), then is $dim(QA) = rank(A)?$ I think ...
0
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1answer
41 views

Why is $Z^n/AZ^n \cong Z/d_1Z\oplus Z/d_2Z..$ where $d_i$ are the entries of normal form of A?

In trying to show that $Card(Z^n/AZ^n)=det(A)$ which has been answered earlier here $\mathrm{card}(\mathbb{Z}^n/M\mathbb{Z}^n) = |\det(M)|$? , the answer refers to the isomorphism $Z^n/AZ^n \cong Z/...
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0answers
47 views

How to show that $\#(\mathbb Z^n/N) = \det(A)$ where $N$ is the subgroup of $\mathbb Z^n$ generated by columns of a square matrix $A$ [duplicate]

$A$ is a $n\times n$ matrix with integer coefficients. I am trying to understand a) and b) claims. a) $Z^n/N$ is finite $\Leftrightarrow$ $det(A)\neq 0$ b) #$(Z^n/N)=|det(A)|$ Looking at the ...
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0answers
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$\mathbb{Z}[1/3]$ is not finitely generated as a $\mathbb{Z}$-module. [duplicate]

I wan't to show that $\mathbb{Z}[1/3]$ is not finitely generated as a $\mathbb{Z}$-module. So I suppose toward a contradiction that it is finitely generated, i.e. $\mathbb{Z}[1/3]=(x_1,...,x_d)$ where ...
1
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1answer
42 views

Finitely generated $k$-algebras beginner examples.

I just found out about finitely generated $k$-algebras (where $k$ is a field). So it is an algebra $A$ for which we have a finite set of elements $(a_1,...,a_n)$ such that every element in $A$ can be ...
1
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1answer
21 views

Distinct terms in Ascending or descending chain of ideals $k-$algebra R which is also a finite dimensional vector space.

Suppose the $k$-algebra $R$ is finite-dimensional as a vector space over $k$, for example, when $R = k[x]/<f(x)>$, where $f$ is any nonzero polynomial in $k[x]$. Then in particular $R$ is a ...
2
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1answer
54 views

On injective module homomorphism $M^m \to M^n$ for a faithful, finitely generated, non-zero module $M$ over a commutative ring

Let $M$ be a non-zero finitely generated faithful module over a commutative ring with unity $R$. If $m,n$ are positive integers such that there exists an injective module homomorphism from $M^m$ to $M^...
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2answers
85 views

Free groups are residually of rank 2

Let $F_X$ denote the free group on the set $X$, and $F_n$ the free group of rank $n$. I have read that any free group is residually $F_2$, and I was trying to understand this. For any free group $F$,...
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1answer
90 views

Determine maximal ideals of $\mathbb R[x]/(x^2)$

Artin Algebra Chapter 11 For (b), a solution can be found here, which I think is the same as Takumi Murayama here. I have a question about the solution of Brian Bi here. What does it mean that "...
2
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0answers
25 views

Index of a common normal core

Let $A, B, C$ be infinite groups and suppose that there are injective homomorphisms $\iota_A \colon C \to A$ and $\iota_B \colon C \to B$ such that $|A : \iota_A(C)|, |B: \iota_B(C)| < \infty$. ...
7
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1answer
97 views

What finite non-abelian group is generated by $\operatorname{diag}(1,w,w^2,\ldots,w^{N-1})$ (with $w=e^{2\pi i/N}$) and a cyclic permutation matrix?

What is the finite nonabelian group? generated by the following elements and satifies the rules: $$A=\left(\begin{array}{ccccc} 1&0&0&\cdots&0\\ 0&\omega&0&\cdots&...
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0answers
11 views

Inversion of Gradient Noise Function

Brief As Possible Given any equation $f(x)$ which adheres to a predefined pattern, there is always a way (sometimes not directly calculable) to get $f^{-1}(x)$ (which may be a set) such that $f^{-1}(...
2
votes
2answers
46 views

Let $A$ be an abelian finitely generated free group and $A/B$ be a torsion group. Show that $rank(A)=rank(B)$.

Let $A$ be an abelian free group that is finitely generated, and let $B\subset A$ be a subgroup of $A$ such that $A/B$ is a torsion group. Show that $rank(A)=rank(B)$. From the hypothesis, I know ...
3
votes
0answers
108 views

Show that $\mathbb{Z} = \langle a, b \mid a^{12} = b,\ ab = ba \rangle$ has dead end elements

This exercise is taken from the book "Office Hours with a Geometric Group Theorist" (Office Hour 15, exercise 8): Exercise: Show that the group $\mathbb{Z}$ has dead end elements with respect to the ...
1
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2answers
50 views

Index of subgroups in residually finite groups

Let $G$ be an infinite finitely generated residually finite group. Is it true that $G$ contains finite-index subgroups of arbitrarily large index? What about the converse: does there exist a finitely ...
0
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0answers
58 views

Localization of finitely generated algebra

Let $R$ be a reduced finitely generated algebra over $\Bbb Z$. Let $T$ be a finite set of prime ideals of $R$. Let $S = \bigcap_{p \not\in T} R \setminus p$. 1) Is it true that $A := S^{-1}R$ is ...