ON AN EMBEDDING PROPERTY OF GENERALIZED CARTER SUBGROUPS

If ℰ and F are saturated formations, ℰ is strongly contained in F (written ℰ F) if: (1.1) For any solvable group G with if ℰ-subgroup E, and F-subgroup F, some conjugate of E is contained in F.This paper is concerned with the problem:(1.2) Given ℰ what saturated formations F satisfy ℰ F The object of this paper is to prove two theorems. The first, Theorem 5.3, shows that if J is a nonempty formation, and ℰ={GGIF(G)∈j}, (F(G) is the Fitting subgroup of G), then any formation F which strongly contains ℰ has essentially the same structure as ℰ in that there is a nonempty formation u such that F = {G /G/F(G) ∈ u}.The second, Theorem 5.8, exhibits a large class of formations which are maximal in the partial ordering In particular, if Ni denotes the formation of groups which have nilpotent length at most i, then Ni is maximal in Since for N= Ni, the N-subgroups of a solvable group G are the Carter subgroups, question (1.2) is settled for the Carter subgroups. © 1969 by Pacific Journal of Mathematics.