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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The Definition of Hyperbolic Triangle Group [closed]

Outer Circles, An Introduction to Hyperbolic $3-$Manifolds. Albert Marden. Page $86$. A Fuchsian group $\Gamma$ is called a (hyperbolic) triangle group of signature $(p,q,r)$, $2 \leq p,q,r \leq \...
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$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 ...
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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 ...
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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\...
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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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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_{...
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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 ...
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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 ...
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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}.....
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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_\...
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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 ...
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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 ...
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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 ...
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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\...
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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 ...
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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 $b$ ...
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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 ...
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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}...
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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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There is no natural isomorphism between torsion functor and identity

Let $\mathcal{C}$ be the full subcategory of $\textbf{Ab}$ whose objects are finitely generated abelian groups. Let $F:\mathcal{C}\rightarrow\mathcal{C}$ be the functor sending $A\in\mathcal{C}$ to $...
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Dumb Question: Is My Group Polycyclic?

I have a group extension $1 \to \mathbb{Z}^2 \to G \to \mathbb{Z}^2 \to 1$ with presentation $G = \langle w_1, w_2, z_1, z_2\ |\ w_1w_2w_1^{-1}w_2^{-1}, w_1z_1w_1^{-1}z_1^{-1}, w_1z_2w_1^{-1}z_2^{-1}, ...
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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 ...
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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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What is the kernel of this map $\Phi: F_2 \to \mathbb{Z}_2 \oplus \mathbb{Z}_3$?

What is the kernel $K\leq F_2 = \langle a,b \rangle$ of this map $\Phi: F_2 \to \mathbb{Z}_2 \oplus \mathbb{Z}_3$ given by $a \mapsto (1+2\mathbb{Z},0+3\mathbb{Z})$ and $b\mapsto (0+2\mathbb{Z}, 1+3\...
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3 votes
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Efficient way to define generating of finite groups

Suppose $(G,\cdot)$ is a group and $X \subset G$ is a nonempty set. The group generated by $X$ is a subgroup of $(G,\cdot)$ with carrier set $$\langle X \rangle = \left\{ g_1^{i_1} \cdot \ldots \cdot ...
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Change of rings of scalars and projecivity?

Let $f:R\to S$ be a ring homomorphism and M be a left S-module. We can consider M as an R-module via $ r.m := f(r)m $. I know that if M is a flat S-module and S is flat as R-module then M is a flat R-...
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2 votes
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$p$-adic topology on a finitely generated module

Suppose that I have a finitely generated $\mathbb Z_p$-module $M$. I have read that there is a "$p$-adic topology" on it. My questions are: I have the product topology on the free group $\...
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Equivalent condition for uniserial torsion module over a PID

This is a homework problem. Suppose, M is a finitely generated torsion module over a PID R. Then, M is uniserial if and only if M has only one elementary divisor. So far I was able to prove the ...
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A characterisation of flatness

A left $R$-module $M$ is flat if and only if every morphism $f:K\to M$ , where $K$ finitely presented, factors through a finitely generated projective left $R$-module. I have found a proof of this ...
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1 answer
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Is the sum of abelian cyclic modules abelian?

Let $m_1R,m_2R,m_3R,\ldots,m_nR$ be cyclic right $R$-module with unity. If every idempotent in the endomorphism ring $\text{End}_R(m_iR)$ is central, then are the idempotents of $\text{End}_R(M)$ ...
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$M$ finitely generated $\nRightarrow$ $N$ and $M/N$ finitely generated

In an algebra lecture we looked at the following lemma: Let $R$ be a ring, $M$ an $R$-module and $N \subset M$ a submodule. Then $N$ and $M/N$ are finitely generated $\implies$ $M$ is also finitely ...
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A group is locally free exactly when its finitely generated subgroups are free

This is Exercise 6.1.9 of Robinson's, "A Course in the Theory of Groups (Second Edition)". According to this search for "locally free finitely generated" in the group theory tag, ...
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Do images and kernels of invariant modules have the form $Mr$ and $\text{Ann}(r)$ respectively for some $r\in R$?

Let $M$ be an $R$-module and $N$ a submodule of $M$. $N$ is said to be a fully invariant submodule of $M$ if $f(N)\subseteq N$ for all $f\in \text{End}_R(M)$. We call an $R$-module $M$ invariant if ...
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Is $\ker(f)$ a finitely generated sub-module of $M$ if $\ker(f)\cong M/f(M),f\in \text{End}_R(M)$?

Let $M_R$ be a module over a ring $R$ with unit and $f\in \text{End}_R(M)$. If $\ker(f)\cong M/f(M)$, then is it right to conclude that $\ker(f)$ is a finitely generated sub-module of $M$? My ...
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1 vote
1 answer
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Finitely generated $R$-module that is not projective or finitely presented

Give an example of a finitely generated $R$-module $M$ (for some commutative ring $R$) that is not projective and is not finitely presented. I was able to find an example of a finitely generated $R$-...
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3 votes
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Two Properties of Perfect Rings

I came across Kunz's theorem about the characterization of regular rings in characteristic $p$. In the paper that I am reading, the author uses perfect rings to prove this result. Perfect rings $R$ ...
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If $A\ast_C B$ is finitely generated, are $A$ and $B$ finitely generated?

I know that if $A$ and $B$ are group of finite rank $n$, there is an amalgamated product $A\ast_{F_2} B$ of rank $2$. It is know that for every $n$ there exist groups $A$ and $B$ of rank $\ge n$ and ...
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If $\langle x \rangle =_{df} H(\{x\})$ with $H(S) , (S $ a set $)$ as defined below , how can $\langle x \rangle$ have more than $3$ elements?

Def. of $H(S) :$ $H(S) = \{g\in G \mid \exists n\in \mathbb N , \exists \{g_1, g_2,... g_n\}\subseteq S\cup S^{-1} , g = g_1 ...g_n\}$. Note : it can be shown that $H(S)$ is simply the same set as $\...
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4 votes
1 answer
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If every finite quotient group of a finitely generated linear group G is solvable, then G is solvable

For this question, I was able to show that each finite quotient is polycyclic: Suppose $N$ is a normal subgroup of finite index. Then, all subgroups of $G/N$ are finite, so $G/N$ is Noetherian. A ...
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2 votes
1 answer
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Simple question about finitely generated algebras

I have two finitely generated algebras $A$ and $B$ over a field $\mathbb{K}$ such that $B\subseteq A$. Is it true that $A=B[a_1,\ldots,a_n]$ for some $a_1,\ldots,a_n\in A$? Motivation: I am trying to ...
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Can $ \frac{\mathbb{Z_{4} \times \mathbb{Z_{6}}}}{\left<(0,2)\right>} $ be isomorphic to $ \mathbb{Z_{8}} $?

This is the example 15.10 of Fraleigh's textbook: Compute the factor group $$ \frac{\mathbb{Z_{4} \times \mathbb{Z_{6}}}}{\left<(0,2)\right>}.$$ $ \mathbb{Z_{4} \times \mathbb{Z_{6}}} $ has ...
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3 votes
3 answers
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Is the cyclic group $\langle x\rangle$ always a subgroup of $G$ for any $x\in G$?

I have been thinking about the following: If $G$ is a finite group and $x\in G$ an element of order $n$ is then $\langle x\rangle$ always a subgroup of $G$? I have the definition that $\langle x\...
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-1 votes
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Versions of Nakayma's Lemma

Nakayama's Lemma: If $M$ is a finitely generated $R$-module and there is an ideal $I\subset R$ with $IM=M$, then there exists $a\in I$ with $am=m$ for all $m\in M$. I'm looking to prove: Let $M$ be a ...
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If $\ker(f_i)\cap f_i(M)=0$ for each $i=1,\cdots, n$, then is it true that ker$(I)\cap I(M)=0$ for any f.g left ideal $I$ of $\text{End}_R(M)$?

Let $R$ be a ring with identity, $M$ a right $R$-module and $S=\text{End}_R(M)$ the ring of $R$-endomorphisms of $M$. Let $I\leq{} _{S}S$ be any nonzero left ideal with a finite number of generators ...
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  • 223
1 vote
2 answers
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Center of finitely generated $C^\ast$-algebra

Let $\mathcal{A}$ be a $C^\ast$-algebra finitely generated by $n$ elements $\{a_1,\dots,a_n\}$. Elements in the center $Z(\mathcal{A})$ commute with each of the generators $$Z(\mathcal{A})\subset\{a\...
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