Finite groups with systems of K-\frakF-subnormal subgroups

2017/05/30 by V. N. Semenchuk, Semenchuk, Vladimir N., Alexander N. Skiba +1
Computer Science · Engineering · Mathematics · #20D10 #20D15 #20D20 #Coding theory and cryptography #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.1705.10476

openalex publication_date 2017/05/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let \frak F be a class of group. A subgroup A of a finite group G is said to be K-\mathfrakF-subnormal in G if there is a subgroup chain A=A₀ ≤ A₁ ≤ ⋯ ≤ A_n=G such that either A_i-1 \trianglelefteq A_i or A_i/(A_i-1)_A_i ∈ \mathfrakF for all i=1, … , n. A formation \frak F is said to be K-lattice provided in every finite group G the set of all its K-\mathfrakF-subnormal subgroups forms a sublattice of the lattice of all subgroups of G. In this paper we consider some new applications of the theory of K-lattice formations. In particular, we prove the following Theorem A. Let \mathfrakF be a hereditary K-lattice saturated formation containing all nilpotent groups. (i) If every \mathfrakF-critical subgroup H of G is K-\mathfrakF-subnormal in G with H/F(H)∈ \mathfrakF, then G/F(G)∈ \mathfrakF. (ii) If every Schmidt subgroup of G is K-\mathfrakF-subnormal in G, then G/G_\mathfrakF is abelian.

Related

• On K\mathfrak F-subnormality and submodularity in a finite group
• On K-ℙ_t-subnormal subgroups of finite groups and related formations
• On some applications of Fitting like subgroups of finite groups
• Joins of σ-subnormal subgroups
• Finite groups with \mathfrakF-subnormal normalizers of Sylow subgroups