Wielandt $\mathfrak{X}$-subgroups

D. O. Revin

Algebra i logika, Tome 63 (2024) no. 3, pp. 301-322

Résumé

Let $\mathfrak{X}$ be a nonempty class of finite groups closed under taking subgroups, homomorphic images, and extensions. We define the concept of a Wielandt $\mathfrak{X}$-subgroup in an arbitrary finite group. It generalizes the concept of a submaximal $\mathfrak{X}$-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 $\mathfrak{X}$-subgroups. Wielandt $\mathfrak{X}$-subgroups possess a number of properties unshareable by submaximal $\mathfrak{X}$-subgroups. There is a hope that, due to these additional properties, the use of Wielandt $\mathfrak{X}$-subgroups will open up new possibilities in realizing Wielandt's program.

Détail

Keywords: Wielandt's program, finite group, maximal $\mathfrak{X}$-subgroup, submaximal $\mathfrak{X}$-subgroup, Wielandt $\mathfrak{X}$-subgroup.