Finite groups with generalized ℙ-subnormal second maximal subgroups

Abstract

A subgroup H of a group G is said to be K-ℙ-subnormal inG [A. F. Vasilyev, T. I. Vasilyeva and V. N. Tyutyanov, On finite groups with almost all K-ℙ-subnormal Sylow subgroups, in Algebra and Combinatorics: Abstracts of Reports of the International Conference on Algebra and Combinatorics on Occasion the 60th Year Anniversary of A. A. Makhnev (Ekaterinburg, 2013), pp. 19–20] if there exists a chain of subgroups H = H0 ≤ H1 ≤ ⋯ ≤ Hn = G such that either Hi-1 is normal in Hi or |Hi : Hi-1| is a prime, for i = 1, …, n. In this paper, we describe finite groups in which every second maximal subgroup is K-ℙ-subnormal.