Every finitely generated abelian group or nilpotent group is polycyclic. This is the fundamental theorem of finitely generated abelian groups. This definition generalizes to direct sums of finitely many abelian groups. Every direct sum of finitely many injective modules is injective. If "G" is abelian, nilpotent, solvable, cyclic or finitely generated, then so is . Module homomorphisms between finitely generated free modules may be represented by matrices. This follows from the classification of finitely generated abelian groups. The structure of finitely generated abelian groups in particular is easily described. Every permutation on finitely many elements can be decomposed into cycles on disjoint orbits. Any commutative ring is the colimit of finitely generated subrings. This is a key step in the classification of finitely generated abelian groups. Theorem&colon; The Cartesian product of finitely many countable sets is countable. The category of finitely generated projective modules over a ring is an exact category. The same definition applies to any topological space which has a finitely generated homology. Then formula 16 can have only finitely many eigenvalues.