▁domain ), ▁and ▁Smith ▁normal ▁form ▁corresponds ▁to ▁class ifying ▁finit ely ▁presented ▁ab elian ▁groups . ▁There ▁is ▁the ▁additional ▁technical ity ▁of ▁showing ▁that ▁a ▁finit ely ▁presented ▁ab elian ▁group ▁is ▁in ▁fact ▁finit ely ▁generated , ▁so ▁Smith ' s ▁classification ▁is ▁not ▁a ▁complete ▁proof ▁for ▁finit ely ▁generated ▁ab elian ▁groups . ▁ ▁The ▁fundamental ▁theorem ▁for ▁finit ely ▁generated ▁ab elian ▁groups ▁was ▁proven ▁by ▁Henri ▁Po inc ar é ▁in ▁, ▁using ▁a ▁matrix ▁proof ▁( which ▁general izes ▁to ▁principal ▁ideal ▁domains ). ▁This ▁was ▁done ▁in ▁the ▁context ▁of ▁computing ▁the ▁hom ology ▁of ▁a ▁complex , ▁specifically ▁the ▁B etti ▁number ▁and ▁t ors ion ▁coefficients ▁of ▁a ▁dimension ▁of ▁the ▁complex , ▁where ▁the ▁B etti ▁number ▁corresponds ▁to ▁the ▁rank ▁of ▁the ▁free ▁part , ▁and ▁the ▁t ors ion ▁coefficients ▁correspond ▁to ▁the ▁t ors ion ▁part . ▁ ▁Kr one cker ' s ▁proof ▁was ▁generalized ▁to ▁finit ely ▁generated ▁ab elian ▁groups ▁by ▁Em my ▁No ether ▁in ▁. ▁ ▁Cor oll aries ▁St ated ▁differently ▁the ▁fundamental ▁theorem ▁says ▁that ▁a ▁finit ely ▁generated ▁ab elian ▁group ▁is ▁the ▁direct ▁sum ▁of ▁a ▁free ▁ab elian ▁group ▁of ▁finite ▁rank ▁and ▁a ▁finite ▁ab elian ▁group , ▁each ▁of ▁those ▁being ▁unique ▁up ▁to ▁isomorphism . ▁The ▁finite ▁ab elian ▁group ▁is ▁just ▁the ▁t ors ion ▁subgroup ▁of ▁G . ▁The ▁rank ▁of ▁G ▁is ▁defined ▁as ▁the ▁rank ▁of ▁the ▁t ors ion -