\({\mathfrak F}\)-bases and subgroup embeddings in finite solvable groups (Q1073177)

Let \({\mathfrak F}\) be a formation of solvable groups locally defined by \(\{\) f(p)\(\}\), where for each prime p, f(p) is a nonempty, subgroup-closed formation contained in \({\mathfrak F}\). Using the notation of \textit{C. J. Graddon} [J. Algebra 18, 574-587 (1971; Zbl 0218.20015)], if G is a solvable group, define \(H\leq G\) to be \({\mathfrak F}\)-pronormal in G if each \({\mathfrak F}\)-basis of G reduces into exactly one conjugate of H in G. Prentice has proved that \(H\leq G\) is \({\mathfrak F}\)-subnormal in G if and only if each \({\mathfrak F}\)-basis of G reduces into every conjugate of H in G. This paper explores the connections between F-pronormality and \({\mathfrak F}\)-subnormality in solvable groups. One result is the following generalization of the Frattini argument: H is \({\mathfrak F}\)-pronormal in G if and only if \(H\leq K\leq L\leq G\) and K \({\mathfrak F}\)-subnormal in L implies \(L=N_ L(H)K\). Another connection investigated is the relationship between subgroups of \(H\leq G\) that are maximal with respect to being \({\mathfrak F}\)-pronormal in G and overgroups of H in G that are minimal with respect to being \({\mathfrak F}\)-subnormal in G.