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If ℰ and ℱ are saturated
formations, ℰ is strongly contained in ℱ (written ℰ≪ℱ) if:
- For any solvable group G with ℰ-subgroup E, and ℱ-subgroup F, some
conjugate of E is contained in F.
This paper is concerned with the problem:
- 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 𝒯 is a nonempty formation, and ℰ = {G∣G∕F(G) ∈𝒯}. (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 = 𝒩1, the 𝒩-subgroups of a solvable
group G are the Carter subgroups, question (1.2) is settled for the Carter
subgroups.
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