▁group . ▁ ▁Every ▁ab elian ▁group , ▁, ▁contains ▁p - basic ▁sub groups ▁for ▁each ▁, ▁and ▁any ▁ 2 ▁p - basic ▁sub groups ▁of ▁ ▁are ▁is omorphic . ▁Ab elian ▁groups ▁that ▁contain ▁a ▁unique ▁p - basic ▁subgroup ▁have ▁been ▁completely ▁character ized . ▁For ▁the ▁case ▁of ▁p - groups ▁they ▁are ▁either ▁divis ible ▁or ▁bounded ; ▁i . e ., ▁have ▁bounded ▁exponent . ▁In ▁general , ▁the ▁isomorphism ▁class ▁of ▁the ▁quot ient , ▁ ▁by ▁a ▁basic ▁subgroup , ▁, ▁may ▁depend ▁on ▁. ▁ ▁General ization ▁to ▁modules ▁▁ ▁The ▁notion ▁of ▁a ▁p - basic ▁subgroup ▁in ▁an ▁ab elian ▁p - group ▁adm its ▁a ▁direct ▁general ization ▁to ▁modules ▁over ▁a ▁principal ▁ideal ▁domain . ▁The ▁existence ▁of ▁such ▁a ▁basic ▁sub module ▁and ▁uniqu eness ▁of ▁its ▁isomorphism ▁type ▁continue ▁to ▁hold . ▁ ▁References ▁▁▁ ▁L ász ló ▁F uchs ▁( 1 9 7 0 ), ▁In finite ▁ab elian ▁groups , ▁Vol . ▁I . ▁P ure ▁and ▁App lied ▁Mathemat ics , ▁Vol . ▁ 3 6 . ▁New ▁York – Lond on : ▁Academ ic ▁Press ▁▁ ▁L . ▁Ya . ▁K ul ik ov , ▁On ▁the ▁theory ▁of ▁ab elian ▁groups ▁of ▁arbitrary ▁cardinal ity ▁( in ▁Russian ), ▁Mat . ▁S b ., ▁ 1 6 ▁( 1 9 4 5 ), ▁ 1 2 9 – 1 6 2 ▁ ▁Category : Ab elian ▁group ▁theory ▁Category : In