On an embedding property of generalized Carter subgroups

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