Two questions in the theory of formations

Abstract

(1) If 9 is a saturated formation, let ,gS denote the formation of all finite soluble groups in which the S-projectors and g-normalizers coincide. By [2], 4.6, the formation gS is saturated if, and only if, gF = MS. In $1 we shall give the canonical local definition of the smallest saturated formation containing dU,. From this we shall deduce that if %F is saturated, 3 necessarily has the form 9 = Y,@(p). (2) A formation 9 is said to have the cover-avoidance property if the S-projectors of each group Q either cover or avoid each chief factor of (li. It was shown in [2], 5.9, that if a saturated formation 9 has the coveravoidance property, then either 9 = %Yz for some set T of primes or 3 = -9$F( p) where S(p) is a formation whose characteristic is the single prime p. We show in $2 that this second possibility can only occur when S(p) = YP , the class of p-groups. Combining these results we obtain the following conclusion: A saturated formation 9 has the cover-avoidance property if, and only if, 9 is either the formation of m-groups for some set z of primes or the formation of p-nilpotent groups for some prime p.