On an embedding property of generalized Carter subgroups

Abstract

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.