On σ-subnormality criteria in finite groups

σ-Subnormality in locally finite groups

“…. X n = G such that, for each 1 i n − 1, X i−1 X i or X i /(X i−1 ) X i is a σ j i -group for some σ j i ∈ σ. Skiba showed that the set of all σ-subnormal subgroups has a strong influence on the structure of a finite soluble group and this led many authors to investigate which of the most relevant theorems about subnormal subgroups have analogs in terms of σ-subnormal subgroups (see for instance [1], [2], [3]). It turns out that one of the main features of σ-subnormal subgroups (in finite groups) is that the join of two σ-subnormal subgroups is always σ-subnormal (see for instance [3]), so they form a sublattice of the lattice of all subgroups.…”

A generalization of Hall's theorem on hypercenter

“…The class of all σ-nilpotent groups is denoted by N σ . This class is a very interesting generalization of the class of nilpotent groups and widely studied (for example, see [3,7,13,15,22,26]). The class N of all nilpotent groups coincides with the class N σ for σ = {{p} | p ∈ P}.…”

A generalisation of Schenkman's theorem

“…In recent years, the investigations of many authors have focussed on the so-called σ-subnormal subgroups (see, for example, [Ski15], [BS17], [Ski18], [Ski19], [Ski20], [GS19], [BBKPAPC20], [BBKPAY20], [KY20], [KT20b]). We explain this notion below.…”