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Full-Text Articles in Physical Sciences and Mathematics
On The Socles Of Fully Invariant Abelian P-Groups, Brendan Goldsmith, P. V. Danchev
On The Socles Of Fully Invariant Abelian P-Groups, Brendan Goldsmith, P. V. Danchev
Articles
The classification of the fully invariant subgroups of a reduced Abelian p-group is a difficult long-standing problem when one moves outside of the class of fully transitive groups. In this work we restrict attention to the socles of fully invariant subgroups and introduce a new class of groups which we term socle-regular groups; this class is shown to be large and strictly contains the class of fully transitive groups. The basic properties of such groups are investigated but it is shown that the classification of even this simplified class of groups, seems extremely difficult.
On The Socles Of Characteristic Subgroups Of Abelian P-Groups, P. V. Danchev, Brendan Goldsmith
On The Socles Of Characteristic Subgroups Of Abelian P-Groups, P. V. Danchev, Brendan Goldsmith
Articles
Fully invariant subgroups of an Abelian p-group have been the object of a good deal of study, while characteristic subgroups have received somewhat less attention. Recently the socles of fully invariant subgroups have been studied and this led to the notion of a socle-regular group. The present work replaces the fully invariant subgroups with characteristic ones and leads in a natural way to the notion of a strongly socle-regular group. A surprising relationship, mirroring that between transitive and fully transitive groups, is obtained.