Wielandt 𝔛 -Subgroups

Let 𝔛 be a nonempty class of finite groups closed under taking subgroups, homomorphic images, and extensions. We define the concept of a Wielandt 𝔛 -subgroup in an arbitrary finite group. It generalizes the concept of a submaximal 𝔛 -subgroup introduced by H. Wielandt and is key in the framework of a program proposed by Wielandt in 1979. One of the central objectives of the program is to overcome difficulties associated with the reduction to factors of a subnormal series within the natural problem of searching for maximal 𝔛 -subgroups. Wielandt 𝔛 -subgroups possess a number of properties unshareable by submaximal 𝔛 -subgroups. There is a hope that, due to these additional properties, the use of Wielandt 𝔛 -subgroups will open up new possibilities in realizing Wielandt’s program.