# 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 center of a finitely generated nilpotent group, finitely generated? [closed]

My professor algebra say that, center of finitely generated nilpotent group is finitely generated!is this true? and if yes, how this is proved?
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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 ...
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### 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 ...
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1 vote
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### Is there any proof of $\#(F/N)=2n$ which doesn't use any group other than $F/N$ itself? (Michael Artin "Algebra 1st Edition")

I am reading "Algebra 1st Edition" by Michael Artin. The following proposition is Proposition (8.3) on p.221 in this book. (8.3) Proposition. The elements $x^n,y^2,xyxy$ form a set of ...
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### Question about a proposition about free groups, generators and relations. Is it true or false that $N=\ker\phi$ holds? Michael Artin "Algebra 1st Ed."

I am reading "Algebra 1st Edition" by Michael Artin. I feel free groups, generators and relations are very difficult. The following proposition is Proposition (8.3) on p.221 in this book. (...
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### Proof that Monomial Ideals are finitely generated by monomials

I'm trying to prove that last part of Lemma 1.2.2 in Sturmfels' "Algorithms in Invariant Theory. For the induction step, we want to prove that n variate monomials M are finitely generated ...
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### Let $B$ be a finite subset of $A$ Show $B$ is in some cyclic subgroup of $A$

Suppose $B$ is a finite subset of $A$. $A =$ {$e^{2πij/n},0 ≤ j < n , n ≥ 1$}. Show $B$ is contained in some cyclic subgroup of $A$. I need to give a good generator of this cyclic subgroup of $A$ ...
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### Normal subgroup of triangle group in GAP

Consider the hyperbolic (extended) triangle group $\Delta(2,3,7)=\langle a,b,c\mid a^2,b^2,c^2,(ab)^2,(bc)^3,(ca)^7\rangle$. I construct it in GAP as a finitely presented group, using the standard ...
1 vote
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### Explicit form of element in a link group

I have a link which is union of knots $K_1\cup\ldots\cup K_n.$ I do know how to find link group $\pi_1(\mathbb{R^3}-K_1\cup\ldots\cup K_{n-1})$, for example, using Wirtinger presentation. What I want ...
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