Equivalence theorems
Two subgroups of an abelian group are equivalent if there is an automorphism of the group mapping one onto the other. The equivalence problem is to find useful necessary and sufficient conditions on the subgroups and their factor groups for equivalence. This paper is a survey of the known results on the equivalence problem, written by the leading contributor to the area. Even for finite \(p\)- groups the problem is not solved, being equivalent to the classification of finite valuated \(p\)-groups. However, solutions are known for several special cases, and these results are what the author calls Equivalence Theorems. The special cases include arbitrary subgroups of a free group, basic subgroups of a countable \(p\)-group, almost balanced subgroups of a simply presented group, isotype subgroups of a simply presented \(p\)-group and various classes of subgroups of a completely decomposable group.
scientific article; zbMATH DE number 218639
scientific article