All groups considered in this paper are finite and soluble. Characterization of Schunck classes and saturated formations by means of certain embedding properties of their associated projectors plays an important part in the Theory of Classes of Groups.
Schunck classes whose projectors are normal subgroups were studied by Blessenohl and Gaschutz. They characterize these classes as the classes
groups, where is a set of primes see 2, III, p. 303 . On the other hand, if F F s S S is the saturated formation of all soluble -groups, then F F-projectors always cover or avoid chief factors. In fact, Doerk and Hawkes give a complete description of the saturated formations for which this property holds. It turns out that F F-projectors are CAP Ž . subgroups subgroups with the cover and avoidance property in every group if and only if either F F s S S for some set of primes, or F F s S S X S S p p for some prime p. The explicit determination of CAP Schunck classes is still an open question. Forster made great progress in this direction. He öbtained the classification of normally embedded Schunck classes. DEFINITIONS. Let U be a subgroup of a group G. Ž . a If p is a prime, we say that U is p-normally embedded in G if a Sylow p-subgroup of U is a Sylow p-subgroup of some normal subgroup of G. Ž . b We say that U is normally embedded in G if U is p-normally embedded in G for all primes p.