Partitions of Finite Abelian Groups

A collection of subgroups Ot. O2, ... , On of a group 0 constitute a partition of 0 if every non-zero element of 0 is in one and only one of the groups 0 1 , O2, ... , On. We shall give conditions on the existence of partitions that consist of ni groups of order qi i = 1, 2, ... ,k and where ql n', d = g.c.d.(n, n') and if the order of the group 0 is p2n then it is easy to prove that