Finitely

Every subgroup and factor group of a finitely generated abelian group is again finitely generated abelian.

Non-finitely generated abelian groups Note that not every abelian group of finite rank is finitely generated; the rank 1 group is one counterexample, and the rank-0 group given by a direct sum of countably infinitely many copies of is another one.

The fundamental theorem of abelian groups states that every finitely generated abelian group is the direct product of finitely many finite primary cyclic and infinite cyclic groups.

They admit a classification of finite p-groups in directed coclass graphs consisting of only finitely many coclass trees whose (infinitely many) members are characterized by finitely many parametrized presentations.

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

A charge is a generalization in both directions: it is a finitely additive, signed measure.

A ring is called universally catenary if any finitely generated algebra over it is catenary.

A similar unsolved (as of 2011) question in free probability theory asks whether the von Neumann group algebras of any two non-abelian finitely generated free groups are isomorphic.

A variant of this construction is the direct sum (also called coproduct and denoted ), where only tuples with finitely many nonzero vectors are allowed.

Because at least one such infinite schema is required (ZFC is not finitely axiomatizable), this shows that the axiom schema of replacement can stand as the only infinite axiom schema in ZFC if desired.

Both these results can be proved by the tube lemma which is used in the proof that a product of finitely many compact spaces is compact.

Bourbaki, "Algebra", III, where the case of finitely generated projective modules is treated.

But if there are only finitely many summands, then the Banach space direct sum is isomorphic to the Hilbert space direct sum, although the norm will be different.

By the fundamental theorem of finitely generated abelian groups it is therefore a finite direct sum of copies of Z and finite cyclic groups.

Compact sets are those that can be covered by finitely many sets of arbitrarily small size.

Contrary to Arrow's impossibility theorem for finitely many individuals, such a rule satisfies the conditions (properties) that Arrow proposes (e.

Dunwoody proved the Wall conjecture for finitely presented groups in 1985.

Events in probability spaces Defining all subsets of the sample space as events works well when there are only finitely many outcomes, but gives rise to problems when the sample space is infinite.

Every subset of a nowhere dense set is nowhere dense, and the union of finitely many nowhere dense sets is nowhere dense.

Examples * The integers are a finitely generated abelian group.