# 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.

365 questions
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### A question regarding “equality” of word lengths for two minimal generating sets of a finite group

Let $G$ be a finite group $d(G) = \min_{<S>=G}|S|$. Suppose that $|X|=|Y|=d(G)$ and $<X>=<Y>=G$. Let $|g|_X$ be the word length of $g$ with respect to $X$ and $|g|_Y$ be the word ...
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### Subgroup of finitely-generated subgroup

Is there a standard name for this concept: Let $H \leq G$ be groups. Say $H$ is ?? if there is a finitely-generated group $K \leq G$ such that $H \leq K$. What should one use in place of "??"? I'm ...
1answer
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### Is a finitely generated module over the field of fractions is also finitely generated over the original integral domain?

Let $R$ be an integral domain and $F$ its field of fractions. Let $M$ be a finitely generated $F$-module. Question: Is $M$ also a finitely generated $R$-module? I know that $M$ is an $R$-module ...
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### correspondence covering spaces of a free group and its finite index subgroups

Let $F$ be a free group. Let's take a finite covering space $\Gamma$ of the graph $\Delta$ representing $F$ (via its fundamental group so $\pi_1(\Delta) \simeq F$ ). In the proof of Marshall Halls's ...
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### Necessary and sufficient conditions that $\langle \zeta, (ij), \lvert\lvert k\, \ell \rvert\rvert, \xi_M\rangle$ generates $\mathscr{P}_n.$

Throughout I use cycle notation and write maps $m:X\to Y$ on the right of their arguments (e.g. $xm=y$ for $m(x)=y$). Let $\zeta=(12\dots n)$. This question is inspired by the following questions: ...
1answer
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### Index of the projection of a subgroup on a quotient by finite normal subgroup.

Given a finitely generated group $G$ and a finite normal subgroup $N \leq G$. I am trying to compare finite index subgroups in $G$ and $G/N$. I know that $H$ is a subgroup of $G$ iff $H/N$ is a ...
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### Uncorrelatedness for random elements of finitely generated groups?

Suppose $G$ is a finitely generated group, $A$ is its finite set of generators. Lets denote the metric induced by the Cayley graph $Cay(G, A)$ on $G$ as $d$. Suppose $\{X_i\}_{n = 0}^\infty$ is a ...
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### Finitely generated nilpotent group is isomorphic to a quotient of the free nilpotent group.

Let $F^{(r)}$ be the free group generated by $r$ elements. Let $\gamma_n(F^{(r)})$ denote its lower central series. Finally, let $F_{n,r} = F^{(r)}/\gamma_{n+1}(F^{(r)})$ be the free nilpotent group ...
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### Finding generators of a group from its action on a topological space

Summary I believe I've written a geometric group theory flavoured proof with a mistake in it, but I'm struggling to see why it might be wrong. I haven't found a counter example, but it also feels too ...
3answers
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### Finitely generated, non abelian, infinite group

I was making a diagram of different types of groups; finite /infinite, cyclic / non-cyclic, finitely generated / inifinitely generated, but realized that I didn't have any examples og infinite groups, ...
1answer
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### Prove or disprove: $S_{10} = \langle (1,3),(1,2, … ,10) \rangle$

Prove or disprove: $S_{10} = \langle (1,3),(1,2, ... ,10) \rangle$ I know that $S_{10}=\langle (1,2) , (1,2,...,10) \rangle$. I tried to use this fact to prove the above but failed. It made me think ...
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### Finding generator of a Schnorr group

A Schnorr group is a large prime-order subgroup of $\Bbb Z^*_p$, the multiplicative group of integers modulo $p$. To generate such a group, we find $p = qr + 1$ such that $p$ and $q$ are prime. ...
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### When are indecomposable projective modules finitely generated?

What conditions to you need to put on your ring to guarantee that the indecomposable projective modules are all finitely generated? Edit: I was hoping there was some general result for this. If your ...
1answer
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