On the nilpotent length of maximal subgroups of finite soluble groups (Q1332492)

Let \(n(G)\) denote the Fitting length of a finite solvable group \(G\). The author relates \(n(G)\) to \(n(U)\) for a maximal subgroup \(U\) of \(G\) by noting that \(n(U) = n(G) - i\) for \(i = 0, 1\) or \(2\). Examples indicate that this is best possible. Next Schunck classes \({\mathfrak N}^ i\) of groups that have nilpotent length at most \(i\) are considered. With the above result, the author proves that a group \(G\) whose \({\mathfrak N}^ i\)- projectors coincide with the \({\mathfrak N}^ i\)-injectors belongs to \({\mathfrak N}^ i\) for \(i \geq 2\). It is known that if \(N\) is a normal nilpotent subgroup of a group \(G\) and \(G/N \in {\mathfrak N}^ i\), the \({\mathfrak N}^ i\)-maximal supplements to \(N\) in \(G\) form a conjugate class. This is needed to prove that the number of conjugate classes of maximal subgroups \(U\) of \(G\) with \(n(U) = n(G) - 2\) is at most \(n(G) - 2\).