On Saturated Formations Which Are Special Relative to the Strong Covering-avoidance Property

Let ? be a saturated formation of finite soluble groups. Let G be a finite soluble group and F an if -projector of G. Then F is said to satisfy the strong covering-avoidance property if (i) F either covers or avoids each chief factor of G, and (ii) FC\L/FC\K is a chief factor of F whenever L/K is a chief factor of G covered by F. Let C?„ denote the class of all finite soluble G in which the y -projectors satisfy the strong covering-avoidance property. <?„ is a formation. Let %tf be the class of groups G in which an if-normalizer is also an if -projector. ïta is a formation studied by Klaus Doerk. Note that î)<r C C<r. 'J is said to be G -special if C„ = Vtf . The purpose of this note is to study C -special formations. Two characterizations of C-special formations are given. Let £ be a positive integer and let Ä denote the class of finite soluble groups G whose Fitting length is at most i. Then Jl is C -special. Finally, the formation (?„ is saturated if and only if if is the class of all finite soluble groups. Let J be a saturated formation which contains the class Jl of finite nilpotent groups as a subformation and let J be a property which a subgroup of a finite group may possess. The class of all finite soluble groups in which the J-projectors have property f is denoted by J« . J is said to be special relative to f or simply J-special if J t, coincides with the formation i)„of all finite soluble groups in which the J-normalizers are also the J'-projectors. Let J be defined as follows: A subgroup H of G is said to have property J if G = CG(R/S)H whenever R/S is a chief factor of G covered by H. One can easily check that in this case J„ = Stf, the class of all finite soluble groups G in which the J-projectors cover only the î-central chief factors of G (see Doerk [2, Definition 4.1 (a)]). In [2], Doerk characterizes Received by the editors September 4, 1973. AMS (MOS) subject classifications (1970). Primary 20D10.