▁refers ▁to ▁any ▁of ▁a ▁number ▁of ▁different ▁mathematical ▁concepts ▁named ▁after ▁N iels ▁Hen rik ▁Ab el : ▁ ▁Group ▁theory ▁ ▁Ab elian ▁group , ▁a ▁group ▁in ▁which ▁the ▁binary ▁operation ▁is ▁commut ative ▁Category ▁of ▁ab elian ▁groups ▁Ab ▁has ▁ab elian ▁groups ▁as ▁objects ▁and ▁group ▁hom omorph isms ▁as ▁morph isms ▁Met abel ian ▁group , ▁a ▁group ▁where ▁the ▁commut ator ▁subgroup ▁is ▁ab elian ▁Ab elian isation ▁ ▁Gal ois ▁theory ▁ ▁Ab elian ▁extension , ▁a ▁field ▁extension ▁for ▁which ▁the ▁associated ▁Gal ois ▁group ▁is ▁ab elian ▁ ▁Real ▁analysis ▁ ▁Ab elian ▁and ▁ta uber ian ▁the or ems , ▁used ▁in ▁the ▁summ ation ▁of ▁diver gent ▁series ▁ ▁Function al ▁analysis ▁ ▁Ab elian ▁von ▁Ne umann ▁algebra , ▁a ▁von ▁Ne umann ▁algebra ▁of ▁operators ▁on ▁a ▁Hil bert ▁space ▁in ▁which ▁all ▁elements ▁comm ute ▁ ▁Top ology ▁and ▁number ▁theory ▁ ▁Ab elian ▁variety , ▁a ▁complex ▁tor us ▁that ▁can ▁be ▁embedded ▁into ▁project ive ▁space ▁Ab elian ▁surface , ▁a ▁two - dimensional ▁ab elian ▁variety ▁Ab elian ▁function , ▁a ▁mer omorphic ▁function ▁on ▁an ▁ab elian ▁variety ▁Ab elian ▁integral , ▁a ▁function ▁related ▁to ▁the ▁in defin ite ▁integral ▁of ▁a ▁differential ▁of ▁the ▁first ▁kind ▁ ▁Category ▁theory ▁ ▁Pre - abel ian ▁category , ▁an ▁add itive ▁category ▁that ▁has ▁all ▁k ern els ▁and ▁c ok ern els ▁Ab elian ▁category , ▁a ▁pre abel ian ▁category ▁in ▁which