On saturated formations which are special relative to the strong covering-avoidance property

Abstract

Let F \mathfrak {F} be a saturated formation of finite soluble groups. Let G G be a finite soluble group and F F an F \mathfrak {F} -projector of G G . Then F F is said to satisfy the strong covering-avoidance property if (i) F F either covers or avoids each chief factor of G G , anaaa F ∩ L / F ∩ K F \cap L/F \cap K is a chief factor of F F whenever L / K L/K is a chief factor of G G coverzd by F F Let C F {\mathcal {C}_\mathfrak {F}} denote the class of all finite soluble G G in which the F \mathfrak {F} -projectors satisfy the strong covering-avoidance property. C F {\mathcal {C}_\mathfrak {F}} is a formation. Let Y F {\mathcal {Y}_\mathfrak {F}} be the class of groups G G in which an F \mathfrak {F} -normalizer is also an F \mathfrak {F} -projector. Y F {\mathcal {Y}_\mathfrak {F}} aa is a formation studied by Klaus Doerk. Note that Y F ⊆ C F {\mathcal {Y}_\mathfrak {F}} \subseteq {\mathcal {C}_\mathfrak {F}} . F \mathfrak {F} is said to be C \mathcal {C} -special if C F = Y F {\mathcal {C}_\mathfrak {F}} = {\mathcal {Y}_\mathfrak {F}} . he puraaaaaaaaaaas note is to sdy C \mathcal {C} -special formations. Two characterizations of C \mathcal {C} -special formations are given. Let i i be a positive integer and let N ( i ) {\mathfrak {N}^{(i)}} denote the class of finite soluble groups G G whose Fitting length is at most i i . Then N ( i ) {\mathfrak {N}^{(i)}} is C \mathcal {C} -special. Finally, the formation C F {\mathcal {C}_\mathfrak {F}} aaaaaturated if and only if F \mathfrak {F} is the class of all finite soluble groups.