If E and F are saturated formations, we say that E is strongly contained in F if for any solvable group G with E-subgroup, E, and F-subgroup, F,some conjugate of E is contained in F.In this paper, we investigate the problem of finding the formations which strongly contain a fixed saturated formation E.
Our main results are restricted to formations, E, such that E = {G|G/F(G) ϵT}, where T is a non-empty formation of solvable groups, and F(G) is the Fitting subgroup of G.If T consists only of the identity, then E=N, the class of nilpotent groups, and for any solvable group, G, the N-subgroups of G are the Carter subgroups of G.
We give a characterization of strong containment which depends only on the formations E, and F.From this characterization, we prove:
If T is a non-empty formation of solvable groups, E = {G|G/F(G) ϵT}, and E is strongly contained in F, then
(1) there is a formation V such that F = {G|G/F(G) ϵV}.
(2) If for each prime p, we assume that T does not contain the class, Sp’, of all solvable p’-groups, then either E = F, or F contains all solvable groups.
This solves the problem for the Carter subgroups.
We prove the following result to show that the hypothesis of (2) is not redundant:
If R = {G|G/F(G) ϵSr’}, then there are infinitely many formations which strongly contain R.
