▁denomin ators ; ▁then ▁ ▁cannot ▁be ▁generated ▁by ▁. ▁The ▁group ▁ ▁of ▁non - zero ▁rational ▁numbers ▁is ▁also ▁not ▁finit ely ▁generated . ▁The ▁groups ▁of ▁real ▁numbers ▁under ▁addition ▁ ▁and ▁non - zero ▁real ▁numbers ▁under ▁multiplication ▁ ▁are ▁also ▁not ▁finit ely ▁generated . ▁ ▁Classification ▁The ▁fundamental ▁theorem ▁of ▁finit ely ▁generated ▁ab elian ▁groups ▁can ▁be ▁stated ▁two ▁ways , ▁general izing ▁the ▁two ▁forms ▁of ▁the ▁fundamental ▁theorem ▁of ▁finite ▁ab elian ▁groups . ▁The ▁theorem , ▁in ▁both ▁forms , ▁in ▁turn ▁general izes ▁to ▁the ▁structure ▁theorem ▁for ▁finit ely ▁generated ▁modules ▁over ▁a ▁principal ▁ideal ▁domain , ▁which ▁in ▁turn ▁adm its ▁further ▁general izations . ▁ ▁Primary ▁decomposition ▁The ▁primary ▁decomposition ▁form ulation ▁states ▁that ▁every ▁finit ely ▁generated ▁ab elian ▁group ▁G ▁is ▁is omorphic ▁to ▁a ▁direct ▁sum ▁of ▁primary ▁cy clic ▁groups ▁and ▁infinite ▁cy clic ▁groups . ▁A ▁primary ▁cy clic ▁group ▁is ▁one ▁whose ▁order ▁is ▁a ▁power ▁of ▁a ▁prime . ▁That ▁is , ▁every ▁finit ely ▁generated ▁ab elian ▁group ▁is ▁is omorphic ▁to ▁a ▁group ▁of ▁the ▁form ▁ ▁where ▁n ▁ ≥ ▁ 0 ▁is ▁the ▁rank , ▁and ▁the ▁numbers ▁q 1 , ▁... , ▁q t ▁are ▁powers ▁of ▁( not ▁necessarily ▁distinct ) ▁prime ▁numbers . ▁In ▁particular , ▁G ▁is ▁finite ▁if ▁and ▁only ▁if ▁n ▁= ▁ 0 . ▁ ▁The ▁values ▁of ▁n , ▁q 1 , ▁... , ▁q t ▁are ▁( up ▁to