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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Finitely generated k-Algebra
If A is a communative, associative k-Algebra, it is finitely generated as an Algebra, if there exists, $a_1,...,a_n$, so that the morphism:
$$
\phi_A: K[X_1,...,X_n] \mapsto A\\
f \to f(a_1,...,a_n)
$$...
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"Abstract" presentation of $SL(2,\mathbb Z)$
I understand that $SL(2,\mathbb Z)$ is finitely generated, and one can exhibit matrices $S$ and $T$ that generate it. It is often stated that there is also an "abstract" presentation
$$
\...
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Is the direct product of finitely-generated groups cancellative? [duplicate]
The direct product is cancellative for finite groups, so I wanted to know if this result holds for finitely-generated groups as well. The proof linked clearly doesn't apply there, but I have been ...
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Canonical form of presentation matrix
Setup: We know that an $R$-module $M$ is finally presented if it is isomorphic to $R^m/AR^n$, where $A$ is an $m$-by-$n$ matrix known as the presentation matrix. Determining $A$ amounts to picking a ...
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Is there a module $M$ over a ring $R$ with unity such that $M = Rx$ and $M = Ry + Rz$ but $Ry \lneq M$ and $Rz \lneq M$, where $x, y, z \in M$?
Let $R$ be a ring with unity.
Is there a module $M$ over $R$ such that $M = Rx$ and $M = Ry + Rz$ but $Ry \neq M$ and $Rz \neq M$, where $x, y, z \in M$? In other words, $Ry$ and $Rz$ are proper ...
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I want to find an example of a finitely generated right module $M$ over $R$ such that $Hom_R(M, R)$ is not a finitely generated left module over $R$ [duplicate]
Let $R$ be a ring with unity. If $M$ is a finitely generated right module over $R$, can we conclude that $\text{Hom}_R(M, R)$ is a finitely generated left module over $R$?
In other words, I want to ...
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If $M$ is a finitely generated right module over $R$, then can we conclude that the canonical map $f: M^{*} \to M^{***}$ is bijective? [duplicate]
Let $R$ be a ring with unity. If $M$ is a right module over $R$, then we can conclude that $M^{*}=\text{Hom}_R(M, R)$ is a left torsionless module over $R$, i.e., the canonical map $f: M^* \to M^{***}$...
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How to prove the free group $F_n$ cannot be generated by $n$ elements as a monoid?
I cannot figure out how to prove that the submonoid of $F_n$ generated by elements $\alpha_1,\dots , \alpha_n$ will never be the whole group.
It clearly is possible to generate the free group on $a_1, ...
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If $M$ is a finitely generated module and $\operatorname{pd}(M) = n < \infty$, then is there a finitely generated projective resolution of $M$?
Let $R$ be a ring with unity, and let $M$ be a finitely generated left module over $R$ with projective dimension $\operatorname{pd}(M)=n<\infty$.
Is there a projective resolution of $M$ of the form
...
- 923
1
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0
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29
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Can we represent finitely generated torsion free $\mathbb Z[t,t^{-1}]$ modules as subgroups of matrices?
According to http://arxiv.org/abs/1006.4153, any finitely generated torsion free $\mathbb Z[t,t^{-1}]$-module can be represented as a quotient of $\oplus_{i\in \mathbb Z} \mathbb Z$.
Proposition 2.4. ...
- 1,641
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Relators in a finitely generated metabelian group
Consider the 2-generated metabelian group $G$ with finite $\mathbf{A}^2$-presentation
$$ \langle\langle a,b \,\vert\, [a,b^{-1}][a^{-1},b][a,b]^2 \rangle\rangle, $$
where $[a,b] = a^{-1}b^{-1}ab$, i.e....
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Finitely Generated Group and Isomorphic Copy of Quotient Groups [duplicate]
Given finitely generated (not necessarily abelian) group $G$ and normal subgroup $N\triangleleft G$, then $G/N\cong H\leq G$. Is this statement correct?
The reason why I am suspicious this to be true ...
- 917
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If every pair of generators generates an elementary group, is the group generated by all the generators necessarily elementary?
Let $G \leq SL(2, \mathbb{C})$. As discussed in Beardon's The Geometry of Discrete Groups (Exercise 5.1, Problem 2), $G$ is elementary (i.e., has a finite $G$-orbit in $\mathbb{R}^3$) if and only if ...
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Is a surjective endomorphism of a finitely generated module over a *non-commutative* ring with unity necessarily an isomorphism?
As is well known, any surjective endomorphism of a finitely generated module $M$ over a commutative ring with unity $R$ must be an isomorphism.
What about the non-commutative case? In other words, is ...
- 923
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3
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Find minimal polynomial over $\mathbb{F}_p$ of a generator of $\mathbb{F}_{p^n}^{*}$
So this is part of an exercise sheet where I thought I had figured it out but turns out I didn't.
Theory from lecture:
We know for a prime $p$ the finite field $\mathbb{F}_{p^n}$ is isomorphic to $\...
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