▁satisfies ▁further ▁technical ▁conditions . ▁This ▁notion ▁was ▁introduced ▁by ▁L . ▁Ya . ▁K ul ik ov ▁( for ▁p - groups ) ▁and ▁by ▁L ász ló ▁F uchs ▁( in ▁general ) ▁in ▁an ▁attempt ▁to ▁form ulate ▁classification ▁theory ▁of ▁infinite ▁ab elian ▁groups ▁that ▁goes ▁beyond ▁the ▁Pr ü fer ▁the or ems . ▁It ▁helps ▁to ▁reduce ▁the ▁classification ▁problem ▁to ▁classification ▁of ▁possible ▁extensions ▁between ▁two ▁well ▁understood ▁classes ▁of ▁ab elian ▁groups : ▁direct ▁sums ▁of ▁cy clic ▁groups ▁and ▁divis ible ▁groups . ▁ ▁Definition ▁and ▁properties ▁▁ ▁A ▁subgroup , ▁, ▁of ▁an ▁ab elian ▁group , ▁, ▁is ▁called ▁ ▁p - basic , ▁for ▁a ▁fixed ▁prime ▁number , ▁, ▁if ▁the ▁following ▁conditions ▁hold : ▁▁▁ ▁is ▁a ▁direct ▁sum ▁of ▁cy clic ▁groups ▁of ▁order ▁ ▁and ▁infinite ▁cy clic ▁groups ; ▁▁ ▁is ▁a ▁p - p ure ▁subgroup ▁of ▁; ▁ ▁The ▁quot ient ▁group , ▁, ▁is ▁a ▁p - div is ible ▁group . ▁ ▁Cond itions ▁ 1 – 3 ▁imply ▁that ▁the ▁subgroup , ▁, ▁is ▁Haus dor ff ▁in ▁the ▁p - adic ▁topology ▁of ▁, ▁which ▁moreover ▁coinc ides ▁with ▁the ▁topology ▁induced ▁from ▁, ▁and ▁that ▁ ▁is ▁dense ▁in ▁. ▁Pick ing ▁a ▁generator ▁in ▁each ▁cy clic ▁direct ▁summ and ▁of ▁ ▁creates ▁a ▁ ▁p - bas is ▁of ▁B , ▁which ▁is ▁analog ous ▁to ▁a ▁basis ▁of ▁a ▁vector ▁space ▁or ▁a ▁free ▁ab elian