A residual generating prefrattini subgroup (Q1085269)

Let G be a finite solvable group, \({\mathcal F}^ a \)nonempty formation and \({\mathcal G}\) the formation of solvable nC-groups. For each formation \({\mathcal F}^ a \)characteristic conjugacy class of subgroups of G, called \(P_{{\mathcal F}}\)-subgroups, is constructed. The \(P_{{\mathcal F}}\)- subgroups have a cover and avoidance property and they exhibit a similar relationship to \({\mathcal F}\) as the prefrattini subgroups have with \({\mathcal G}\). The prefrattini subgroups appear as a special case when \({\mathcal F}={\mathcal G}\). A number of results about the connections between the formation \({\mathcal F}\), the \({\mathcal F}\)-residual and the \(P_{{\mathcal F}}\)- subgroups are obtained. In particular, whether or not a formation is saturated is determined by the structure of the \(P_{{\mathcal F}}\)- subgroups. Also if \(1=K_ 0<K_ 1<...<K_ n=G\) is a Fitting chain for G, then \(G\in {\mathcal F}\) if and only if for each i, the core in \(G/K_ i\) of each \(P_{{\mathcal F}}\)-subgroup of \(G/K_ i\) is the identity subgroup. Let \(P_{{\mathcal F}}\) denote a \(P_{{\mathcal F}}\)-subgroup of G. A saturated formation \({\mathcal H}\) is defined from \({\mathcal F}\), so that there exists a chain of subgroups \(D=G_ 0<G_ 1<...<G_ n=G\), from G to an \({\mathcal H}\)-normalizer D of G. Let core \(P_ i\) denote the core in \(G_ i\) of a \(P_{{\mathcal F}}\)-subgroup of \(G_ i\). It is proved that \(P_{{\mathcal F}}\) is the product of the elements of the set \(\{\) core \(P_ i|\) \(i=0,1,2,...,n\}\). If W is a prefrattini subgroup of G, and \({\mathcal H}\) and the chain of subgroups are defined through \({\mathcal G}\), the above result yields \(W=\Phi (G_ 0)\Phi (G_ 1)...\Phi (G)\).