free ▁part ▁of ▁G ; ▁this ▁is ▁just ▁the ▁number ▁n ▁in ▁the ▁above ▁formulas . ▁ ▁A ▁cor ollary ▁to ▁the ▁fundamental ▁theorem ▁is ▁that ▁every ▁finit ely ▁generated ▁t ors ion - free ▁ab elian ▁group ▁is ▁free ▁ab elian . ▁The ▁finit ely ▁generated ▁condition ▁is ▁essential ▁here : ▁ ▁is ▁t ors ion - free ▁but ▁not ▁free ▁ab elian . ▁ ▁Every ▁subgroup ▁and ▁factor ▁group ▁of ▁a ▁finit ely ▁generated ▁ab elian ▁group ▁is ▁again ▁finit ely ▁generated ▁ab elian . ▁The ▁finit ely ▁generated ▁ab elian ▁groups , ▁together ▁with ▁the ▁group ▁hom omorph isms , ▁form ▁an ▁ab elian ▁category ▁which ▁is ▁a ▁Ser re ▁sub category ▁of ▁the ▁category ▁of ▁ab elian ▁groups . ▁ ▁Non - f initely ▁generated ▁ab elian ▁groups ▁Note ▁that ▁not ▁every ▁ab elian ▁group ▁of ▁finite ▁rank ▁is ▁finit ely ▁generated ; ▁the ▁rank ▁ 1 ▁group ▁ ▁is ▁one ▁counter example , ▁and ▁the ▁rank - 0 ▁group ▁given ▁by ▁a ▁direct ▁sum ▁of ▁count ably ▁infinitely ▁many ▁copies ▁of ▁ ▁is ▁another ▁one . ▁ ▁See ▁also ▁ ▁The ▁Jordan – H öld er ▁theorem ▁is ▁a ▁non - abel ian ▁general ization ▁ ▁Notes ▁ ▁References ▁▁▁ ▁Re print ed ▁( pp . ▁ 3 6 7 – 4 0 9 ) ▁in ▁The ▁Collect ed ▁Mathemat ical ▁Pap ers ▁of ▁Henry ▁John ▁Stephen ▁Smith , ▁Vol . ▁I , ▁edited ▁by ▁J . ▁W . ▁L . ▁Gla isher . ▁Oxford : ▁Clar end