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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.
Goldsmith, B. & Danchev, P. On the socles of characteristic subgroups of Abelian p-groups. Journal of Algebra, vol. 323), pg. 3020-3028. doi:10.1016/j.jalgebra.2009.12.005