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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.

For questions regarding finitely generated groups and other algebraic structures. A structure is called finitely generated if there exists a finite subset that generates it.

The structure of finitely generated abelian groups in particular is easily described. Many theorems that are true for finitely generated groups fail for groups in general. It has been proven that if a finite group is generated by a subset S, then each group element may be expressed as a word from the alphabet S of length less than or equal to the order of the group.

789 questions
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Prove that $\langle x,y,|x^2,y^2\rangle$ is an infinite group

Here is my attempt: $xyx \neq xyxy \neq xyxyx \neq xyxyxy....$ gives infinite many elements in the group. Is this correct?
user614287
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Proalgebraic completion.

For a finitely generated group, say $\Gamma$, what is the meant by of the proalgebraic completion of $\Gamma$? I came across this while seeing a paper on Representation Growth for Linear Groups by Larsen and Lubotzky.
Uday Bhaskar
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Determine if quotient(factor) ring $K[x_{1},\ldots ,x_{n}]/I$ is finitely generated

$K[x_{1},\ldots ,x_{n}]$ is polynomial ring. $I = (f_{1},\ldots ,f_{m})$ - it's ideal, where $(f_{1},\ldots ,f_{n})$ is finite set of polinomials. Task is to programmically determine if quotient(factor) ring over this ideal $K[x_{1},\ldots…
user745
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