On K\mathfrak F-subnormality and submodularity in a finite group

2023/06/21 by В. С. Монахов, Monakhov, Victor S., Irina L. Sokhor +1
Mathematics · #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR)

paper · pdf · doi:10.48550/arxiv.2306.12035

openalex publication_date 2023/06/21 · openalex created_date 2023/06/24 · openalex updated_date 2026/07/28

Abstract

Let \mathfrak F be a formation and let G be a group. A subgroup H of G is K\mathfrak F-subnormal (submodular) in G if there is a subgroup chain H=H₀≤ H₁ ≤ … ≤ H_i ≤ H_i+1≤ … ≤ H_n=G such that for every i either H_i is normal in H_i+1 or H_i+1^\mathfrakF ≤ H_i (H_i is a modular subgroup of H_i+1, respectively). We prove that a primary subgroup R of a group G is submodular in G if and only if R is K\mathfrak U₁-subnormal in G. Here \mathfrakU₁ is the class of all supersolvable groups of square-free exponent. In addition, for a solvable subgroup-closed formation \mathfrakF, every solvable K\mathfrakF-subnormal subgroup of a group G is contained in the solvable radical of G.

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