# 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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### On 2-generated profinite abelian groups

It's true that a 2-generated profinite abelian group is an epimorphic image of $\hat{\mathbb{Z}}\bigoplus\hat{\mathbb{Z}}$? If yes how can I show this?
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### Whether a group containing a free group is also a free group

Suppose $G$ is a group generated by two elements $s,t$. Suppose $H$ is a subgroup of $G$ such that $H= \langle s^k,t^k \rangle$ is a free group where $k$ is some integer not equal to $1$. Does it ...
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### Isomorphism of finitely generated modules over a PID

Review question from my first course on Algebraic Structures: Let $M, N, P$ be finitely generated modules over a PID. a) Show that if $M⊕M≅N⊕N$, then $M≅N$. b) Show that if $M⊕P≅N⊕P$, then $M≅N$. ...
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### Infinite subgroups of $\text{PGL}(2,\mathbb C)$ where each element has finite order

I am interested in finite subgroups of $\text{PGL}(2,\mathbb C)\cong\text{Aut}\,\mathbb C(x)$. Let $G\subset\text{PGL}(2,\mathbb C)$ be finitely generated such that each $g\in G$ has finite order. Is ...
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### Finitely generated $R$-module as a sum of cyclic submodules

I know that $M$ is finitely generated if there exist $a_1, ..., a_n$ in $M$ such that for any $x$ in $M$, there exist $r_1, ..., r_n$ in $R$ such that $x = r_1a_1 + \dots + r_na_n.$ But if I want to ...
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### Different Definition of Fundamental Theorem of Finitely Generated Abelian Groups

I was reading about abelian groups but confused in Fundamental Theorem of Finitely Generated Abelian Groups, so I searched some sources and become even more confused due to the different types of ...
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### A group such that all its subgroups are finitely generated

I have been thinking about the following question: Let $G$ be an infinite group with the property that every proper subgroup of $G$ is finitely generated. Can we say that $G$ is always finitely ...
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### Basic Property of Finite Presentation of Modules

I'm struggling to prove some rudimentary facts about finite presentations of a module $A$ over a ring $\Lambda$, which is defined in Hilton/Stammbach's A Course in Homological Algebra as a short exact ...
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### using finitely generated in proof.

I am trying to show that:Let $R$ be commutative with unity. if $N$ be a finitely generated $R-$module, then the following conditions are equivalent: (a) $N$ is a sum of simple modules. (b) $N$ is a ...
Given a strictly descending chain of finitely generated groups $$H_1 > H_2 > H_3 > \cdots,$$ is it always the case that $H = \cap_{i=1}^{\infty} H_i$ is finitely generated? I think the ...