# 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 ...
1 vote
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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 $\sum_{i=1}^{n}\lambda_{... 1 vote 0 answers 40 views ### Proof that strictly ascending chains of subgroups of finitely generated groups are finite [closed] Question I am looking for a proof that: "any strictly ascending chain of subgroups of a finitely generated group$G$whose union is the whole group$G$must be finite" Attempt I was trying ... 1 vote 1 answer 32 views ### Generating the algebra$\mathcal{O}[\mathbf{GL}(n,\mathbb{C})]$(regular functions) I recently asked a question about the regular functions in$\mathbf{GL}(n,\mathbb{C})$but now that I have read the appendix I am again confused; here is my past question: Understanding regular ... 0 votes 1 answer 35 views ### Order of Finitely Generated Group of Prime Exponent Consider the finitely generated group$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}..... 0 votes 1 answer 44 views ### The intersection of normal subgroups of index n is of finite index Let G be a finitely generated group. n>0 is a fixed integer. \{K_\alpha\} is the set of all normal subgroups of G with index n, that is [G:K_\alpha]=n. Consider \bigcap_\alpha K_\... 1 vote 0 answers 57 views ### If K\rtimes \mathbb{Z} is a finitely generated group but K isn't, must the fixed points of 1_\mathbb{Z} be a finitely generated group? I recently answered the following question: If a finitely generated semi-direct product \mathbb{Z} acting on a non finitely generated group, can there be fixed points? I have a related question: Is ... 1 vote 1 answer 78 views ### If a finitely generated semi-direct product \mathbb{Z} acting on a non finitely generated group, can there be fixed points? Suppose that we have a group G = K\rtimes\mathbb{Z}, where G is finitely generated, but K is not finitely generated, and let \phi(1) be the automorphism of K corresponds to 1_\mathbb{Z} \in ... 0 votes 1 answer 33 views ### Why are the direct summands obtained in the structure theorem for finitely generated modules over a PID indecomposable? The structure theorem tells us that a finitely generated module M over a principal ideal domain R is isomorphic to a direct sum \bigoplus _{i}R/(q_{i}) where ( q_i ) ≠ R (q_{i})\neq R and ... 1 vote 0 answers 86 views ### Suppose G is group of order p^k for p prime and k positive integer. Prove G must contain an element with an order of p Suppose G is group of order p^k for p prime and k positive integer. Prove G must contain an element with an order of p. I have proven that every element of G must be of order p^j, 0\... 2 votes 1 answer 72 views ### Minimal generating set in case of finite non-abelian groups Finding a minimal generating set of a finite group is difficult but in the case of abelian groups using the fundamental theorem of finite abelian groups, we can find it easily. Moreover, corresponding ... 2 votes 1 answer 47 views ### What is the most concise way to present a particular subgroup of F_2? Let A\leq F_2 where F_2 is the free group on \{a,b\}. Assume A is generated by words on a,b such that the total powers of a are 3x for some x\in \mathbb{Z} and the total powers of b ... 2 votes 1 answer 90 views ### Definition of finitely generated module versus finite type in category R-Mod We say an object X in category C=R-Mod is of finite type if for any functor F: I \rightarrow C with I a directed poset, the natural map$$\underrightarrow{\lim} Hom_{\mathcal{C}}(X,F(i))\to ... -1 votes 2 answers 42 views ### Prove finiteness of quotient group [closed] Let$G$be a finitely generated abelian group. Prove that the quotient group$G/2G$is finite. I tried two approaches but did not succeed. Structure Theorem:$G\cong\mathbb{Z}^r\times\prod\mathbb{Z}... 44 views

### 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
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 ...