On $ \sigma $-subnormal subgroups and products of finite groups.

Suppose that σ = { σ i : i ∈ I } is a partition of the set P of all primes. A subgroup A of a finite group G is said to be σ - subnormal in G if A can be joined to G by a chain of subgroups A = A 0 ⊆ A 1 ⊆ ⋯ ⊆ A n = G such that either A i − 1 normal in A i or A i / C o r e A i (A i − 1) is a σ j -group for some j ∈ I , for every 1 ≤ i ≤ n. A σ -subnormality criterion related to products of subgroups of finite σ -soluble groups is proved in the paper. As a consequence, a characterisation of the σ -Fitting subgroup of a finite σ -soluble group naturally emerges. [ABSTRACT FROM AUTHOR]