On ?-subnormal subgroups and products of finite groups

Suppose that o-= {o-i: i E I) is a partition of the set P of all primes. A subgroup A of a finite group G is said to be o --subnormal in G if A can be joined to G by a chain of subgroups A = A0 c A1 c middot middot middot c An = G such that either Ai-1 normal in Ai or Ai/CoreAi(Ai-1) is a o -j-group for some j E I, for every 1 < i < n. A o --subnormality criterion related to products of subgroups of finite o --soluble groups is proved in the paper. As a consequence, a characterisation of the o --Fitting subgroup of a finite o --soluble group naturally emerges.