On the Lattice of II-Subnormal Subgroups of A Finite Group

Abstract

Let σ = {σi | i ∈ I} be a partition of the set of all primes P. Let σ0 ∈ Π ⊆ σ and let I be a class of finite σ0-groups which is closed under extensions, epimorphic images and subgroups. We say that a finite group G is ΠI-primary provided G is either an I-group or a σi-group for some σi ∈ Π \ {σ0} and we say that a subgroup A of an arbitrary group G∗ is ΠI-subnormal in G∗ if there is a subgroup chain A = A0 ≤ A1 ≤ · · · ≤ At = G∗ such that either Ai−1 E Ai or Ai/(Ai−1)Ai is ΠI-primary for all i = 1, . . . , t. We prove that the set LΠI(G) of all ΠI-subnormal subgroups of G forms a sublattice of the lattice of all subgroups of G and we describe the conditions under which the lattice LΠI(G) is modular.