On the socles of characteristic subgroups of Abelian \(p\)-groups.
Recall that a \(p\)-group is said to be `socle regular' if for all fully invariant subgroups \(F\) of \(G\), there exists an ordinal \(\alpha\) such that \(F[p]=(p^\alpha G)[p]\). The socle regular groups were studied by the two authors in earlier work and it was shown that the class of socle-regular groups strictly contains the class of fully transitive groups. Thus this concept is closely related to the concept of (full-) transitivity introduced by Kaplansky. While the fully invariant subgroups have attracted much attention, the characteristic subgroups have been investigated only little. In this paper the authors introduce the stronger concept of `strongly socle-regular' meaning that for all characteristic subgroups \(C\) of \(G\), there exists an ordinal \(\alpha\) such that \(C[p]=(p^\alpha G)[p]\). Many nice properties of strongly socle-regular groups are investigated and the relation to the concepts of socle-regularity and (full-) transitivity is clarified. It is very interesting to see that some of the relationships mirror the relationship between fully transitive groups and transitive groups. Hence the new notions of socle-regular and strongly socle-regular groups can be seen as generalizations of Kaplansky's notions.
scientific article; zbMATH DE number 5718195