A generalization of a Hall theorem

Let [Formula: see text] be some partition of the set [Formula: see text] of all primes, that is, [Formula: see text] and [Formula: see text] for all [Formula: see text]. We say that a finite group [Formula: see text] is [Formula: see text]-soluble if every chief factor [Formula: see text] of [Formula: see text] is a [Formula: see text]-group for some [Formula: see text]. We give some characterizations of finite [Formula: see text]-soluble...