Generalizing quasinormality

Abstract

‎Quasinormal subgroups have been studied for nearly 80 years‎. ‎In finite groups‎, ‎questions concerning them invariably reduce to $p$-groups‎, ‎and here they have the added interest of being invariant under projectivities‎, ‎unlike normal subgroups‎. ‎However‎, ‎it has been shown recently that certain groups‎, ‎constructed by Berger and Gross in 1982‎, ‎of an important universal nature with regard to the existence of core-free quasinormal subgroups generally‎, ‎have remarkably few such subgroups‎. ‎Therefore in order to overcome this misfortune‎, ‎a generalization of the concept of quasinormality will be defined‎. ‎It could be the beginning of a lengthy undertaking‎. ‎But some of the initial findings are encouraging‎, ‎in particular the fact that this larger class of subgroups also remains invariant under projectivities of finite $p$-groups‎, ‎thus connecting group and subgroup lattice structures‎.