Maximal supersoluble subgroups of symmetric groups (Q1191346)

All maximal supersoluble subgroups of symmetric groups are classified. But, in fact, the main result in this paper is the classification of the maximal supersoluble transitive subgroups of the symmetric group \(S_ n\) on \(n\) letters. The proof of the main result splits into two parts. In Section 4 the authors present a certain general construction of maximal supersoluble transitive subgroups of \(S_ n\) and in Section 5 it is shown that there are no others, that is, any maximal supersoluble transitive subgroup of \(S_ n\) is conjugate to one of the subgroups presented in Section 4. Then, the intransitive case can be dealt with by a straightforward reduction to the transitive case to yield the following theorem (cf. Section 7): Let \(n = n_ 1 + \dots + n_ b\) with \(b > 1\) and let \(H = H_ 1 \times H_ 2 \times \dots \times H_ b\), where \(H_ i\) is a maximal supersoluble transitive subgroup of \(S_{n_ i}\); furthermore, set \(\varphi(x) = \prod^ b_{i = 1} (x - n_ i)\). Then \(H\) is a maximal supersoluble subgroup of \(S_ n\) if and only if \((x- 1)(x-2)\nmid \varphi(x)\) and \((x-2^ f)^ 2 \nmid \varphi(x)\) for all \(f\geq 0\). The authors also obtain several other results which are not mentioned here due to the limited space. Suprunenko proved that every symmetric group \(S_ n\) possesses a unique conjugacy class of maximal nilpotent transitive subgroups. This result is important for the work reviewed.