Overgroups of primitive groups. II.
Assume that \(\Omega\) is a set of finite order \(n\), \(G\in\{\text{Sym}(\Omega),\text{Alt}(\Omega)\}\), and \(H\) is a subgroup of \(G\) acting primitively on \(\Omega\). In this paper it is continued the study of the set \(\mathcal O_G(H)\) of overgroups of \(H\) in \(G\) [for part I cf. J. Aust. Math. Soc. 87, No. 1, 37-82 (2009; Zbl 1190.20012)]. Let \(\Lambda\) be a finite lattice. Then \(\Lambda\) has a greatest member \(\infty\) and a least member \(0\). Set \(\Lambda'=\Lambda-\{0,\infty\}\). Regard \(\Lambda\) as a graph with adjacency relation the comparability relation, and define \(\Lambda\) to be disconnected if the graph \(\Lambda'\) is disconnected. Write \(\Delta(m)\) for the lattice of subsets of an \(m\)-set, partially ordered by inclusion. Define a \(D\Delta\)-lattice to be a lattice \(\Lambda\) such that \(\Lambda'\) has \(r>1\) connected components \(\Lambda_i'\simeq\Delta(m_i)\) for some \(m_i>2\). Aschbacher and Shareshian conjecture that no \(D\Delta\)-lattice is an interval in the subgroup lattice of a finite group and Aschbacher reduced the verification of this conjecture to two problems about the overgroup lattices in almost simple groups. The first problem is to prove that if \(X\) is an almost simple group and \(Y\leq X\) then \(\mathcal O_X(Y)\) is not a \(D\Delta\)-lattice. This is accomplished, when \(X\) is an alternating or symmetric group and \(Y\) is primitive (Theorem E).
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