On the structure of abelian $p$-groups

Abstract

Abstract.A new kind of abelian p-gioup, called an A -group, is introduced. This class contains the totally projective groups and Warfield's S-groups as special cases. It also contains the V-groups recently classified by the author. These more general groups are classified by cardinal (numerical) invariants which include, but are not limited to, the Ulm-Kaplansky invariants. Thus the existing theory, as well as the classification, of certain abelian ^-groups is once again generalized.Having classified /(-groups (by means of a uniqueness and corresponding existence theorem) we can successfully study their structure and special properties. Such a study is initiated in the last section of the paper.

Characteristic properties of large subgroups in primary abelian groups

“…p-torsion A-groups. Several details on A-groups appear in [7]. For example, any ptorsion abelian A-group is an isotype subgroup of a totally projective p-group with special properties described in [7].…”

“…Thus L 1 is an A-group and L/L 1 is a direct sum of cyclics ( [6], p. 110, Theorem 18.1 of L Kulikov). Finally, again using [7], L must be an A-group. Conversely, if L is an A-group, then so is L 1 = G 1 by making use of [1] and [7].…”

“…Finally, again using [7], L must be an A-group. Conversely, if L is an A-group, then so is L 1 = G 1 by making use of [1] and [7]. On the other hand, L/L 1 is a direct sum of cyclics (see [7]) and also it is a large subgroup of G/G 1 owing to [1].…”

“…Conversely, if L is an A-group, then so is L 1 = G 1 by making use of [1] and [7]. On the other hand, L/L 1 is a direct sum of cyclics (see [7]) and also it is a large subgroup of G/G 1 owing to [1]. Thus we have in virtue of [1] that G/G 1 is a direct sum of cyclics, and finally [7] is applicable to get that G is an A-group.…”

“…Here we quote some major problems that remain unsolved: Is it true that G is a (1) direct sum of σ -summable p-groups; (2) p ω+n projective p-group; (3) thick p-group; (4) essentially finitely indecomposable p-group; (5) pure-complete p-group; (6) semi-complete pgroup; (7) direct sum of torsion-complete p-groups ([1], Problem 4); (8) Fuchs 5 p-group; (9) Q-p-group; (10) ℵ 1 -separable p-group; (11) weakly ℵ 1 -separable p-group if and only if so is L? However, not all properties of G are inherited for L and conversely.…”