The fundamental theorem of finite abelian groups states that every finite abelian group formula_49 can be expressed as the direct sum of cyclic subgroups of prime-power order; it is also known as the basis theorem for finite abelian groups. Moreover, automorphism groups of cyclic groups are examples of abelian groups. This is generalized by the fundamental theorem of finitely generated abelian groups, with finite groups being the special case when "G" has zero rank; this in turn admits numerous further generalizations.