▁re arr anging ▁the ▁indices ) ▁uniqu ely ▁determined ▁by ▁G , ▁that ▁is , ▁there ▁is ▁one ▁and ▁only ▁one ▁way ▁to ▁represent ▁G ▁as ▁such ▁a ▁decomposition . ▁ ▁In variant ▁factor ▁decomposition ▁We ▁can ▁also ▁write ▁any ▁finit ely ▁generated ▁ab elian ▁group ▁G ▁as ▁a ▁direct ▁sum ▁of ▁the ▁form ▁ ▁where ▁k 1 ▁div ides ▁k 2 , ▁which ▁div ides ▁k 3 ▁and ▁so ▁on ▁up ▁to ▁k u . ▁ ▁Again , ▁the ▁rank ▁n ▁and ▁the ▁invariant ▁factors ▁k 1 , ▁... , ▁k u ▁are ▁uniqu ely ▁determined ▁by ▁G ▁( here ▁with ▁a ▁unique ▁order ). ▁ ▁The ▁rank ▁and ▁the ▁sequence ▁of ▁invariant ▁factors ▁determine ▁the ▁group ▁up ▁to ▁isomorphism . ▁ ▁Equ ival ence ▁These ▁statements ▁are ▁equivalent ▁as ▁a ▁result ▁of ▁the ▁Chinese ▁remainder ▁theorem , ▁which ▁implies ▁that ▁ ▁if ▁and ▁only ▁if ▁j ▁and ▁k ▁are ▁cop r ime . ▁ ▁History ▁The ▁history ▁and ▁credit ▁for ▁the ▁fundamental ▁theorem ▁is ▁complicated ▁by ▁the ▁fact ▁that ▁it ▁was ▁proven ▁when ▁group ▁theory ▁was ▁not ▁well - est ab lished , ▁and ▁thus ▁early ▁forms , ▁while ▁essentially ▁the ▁modern ▁result ▁and ▁proof , ▁are ▁often ▁stated ▁for ▁a ▁specific ▁case . ▁Br ief ly , ▁an ▁early ▁form ▁of ▁the ▁finite ▁case ▁was ▁proven ▁in ▁, ▁the ▁finite ▁case ▁was ▁proven ▁in ▁, ▁and ▁stated ▁in ▁group - the or etic ▁terms ▁in ▁. ▁The ▁finit ely ▁presented ▁case ▁is ▁solved ▁by ▁Smith ▁normal ▁form , ▁and ▁hence