On the Kegel-Wielandt σ-problem for binary partitions

Abstract

Let σ = {σi ∶ i ∈ I} be a partition of the set ℙ of all prime numbers. A subgroup X of a finite group G is called σ-subnormal in G if there is a chain of subgroups X = X₀ ⊆ X₁ ⊆ ... ⊆ Xn = G where for every j = 1, … , n the subgroup Xj₋₁ is normal in Xj or Xj ∕ Core Xj (Xj₋₁) is a σi-group for some i ∈ I. In the special case that σ is the partition of ℙ into sets containing exactly one prime each, the σ-subnormality reduces to the familiar case of subnormality. A fnite group G is σ-complete if G possesses at least one Hall σi-subgroup for every i ∈ I, and a subgroup H of G is said to be σi-subnormal in G if H ∩ S is a Hall σi-subgroup of H for any Hall σi-subgroup S of G. Skiba proposes in the Kourovka Notebook the following problem (Question 19.86), that is called the Kegel–Wielandt σ-problem: Is it true that a subgroup H of a σ-complete group G is σ-subnormal in G if H is σi-subnormal in G for all i ∈ I? The main goal of this paper is to solve the Kegel–Wielandt σ-problem for binary partitions.