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finitely generated AGG

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finitely generated MAT

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It is a perfectly good theory with many good properties: it is complete, decidable, finitely axiomatizable, and so on. en.wikipedia.org

In particular, this means a finitely generated 3-manifold group is finitely presentable. en.wikipedia.org

A basic question in this direction is: are there finitely or infinitely many rational points on a given curve (or surface)? en.wikipedia.org

Any noetherian scheme can be written uniquely as the union of finitely many maximal irreducible non-empty closed subsets, called its irreducible components. en.wikipedia.org

Finite geometry itself, the study of spaces with only finitely many points, found applications in coding theory and cryptography. en.wikipedia.org

Because the actual sequence and the representative sequence are in the same equivalence class, their entries are the same after finitely many prisoners. en.wikipedia.org

Even very simple flows, such as the blinking vortex, or finitely resolved wind fields can generate exceptionally complex patterns from initially simple tracer fields. en.wikipedia.org

It is particularly useful where finiteness assumptions are satisfied, for example finitely generated groups, or finitely presented groups (i.e. in addition the relations are finite). en.wikipedia.org

Every finitely generated abelian group or nilpotent group is polycyclic. en.wikipedia.org

There is a natural similar question for iterating splittings of finitely generated groups over finite subgroups. en.wikipedia.org

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