The following question was suggested by a theorem of Pálfy and Pudlak: is each nonempty finite lattice isomorphic to an overgroup lattice \(O_G(H)\) for some finite group \(G\) and subgroup \(H\) of \(G\)? The answer is almost certainly negative. However, the question has remained open for almost 30 years.
For a finite lattice \(L\), let \(L'\) be the poset obtained by removing the minimum and maximum elements of \(L\) and, for \(n\in\mathbb N\), let \(\Delta(n)\) be the lattice of all subsets of \(\{1,\dots,n\}\), ordered by inclusion. For positive integers \(m_1,\dots,m_t\), the lattice \(D\Delta(m_1,\dots,m_t)\) is defined as follows: \(D\Delta(m_1,\dots,m_t)'\) is a disjoint union of posets \(C_1,\dots,C_t\) with \(C_i\cong\Delta(m_i)'\) for all \(i\) and if \(i\neq j\) then no element of \(C_i\) is comparable to any element of \(C_j\).
Shareshian advanced the conjecture that if \(t>1\) and \(m_1\geq m_2\geq\cdots\geq m_t\geq 3\), then there does not exist a finite group \(G\) and subgroup \(H\) of \(G\) such that the overgroup lattice \(O_G(H)\) is isomorphic to \(D\Delta(m_1,\dots,m_t)\). The present paper is one step in a program to prove this conjecture.
In a previous paper the author defined the notion of a “signalizer lattice” and showed that if a lattice \(\Lambda\) in the class \(D\Delta\) is a finite subgroup interval lattice, then there exists an almost simple group \(G\) such that either \(\Lambda\cong O_G(H)\) for some \(H\leq G\) or there exists a finite non-Abelian simple group \(L\) and a pair \(\gamma=(H,J)\) with \(J\trianglelefteq H\leq G\) such that \(F^*(H/J)\cong L\), \(G=\langle W_0(\gamma),H\rangle\) and \(\Lambda\) is isomorphic to the lower signalizer lattice \(\Xi(\gamma)\) (here \(W_0(\gamma)\) is the set of the \(H\)-invariant subgroups \(W\) of \(G\) such that \(H\cap W=J\) and \(W\leq F^*(G)J\) while \(\Xi(\gamma)\) is the lattice obtained by adjoining a greatest element to the poset \(W_0(\gamma)\)).
This reduces the study of Shareshian’s conjecture to two questions about sublattices of the lattice of subgroups of almost simple groups. The obvious first test cases for the two questions are the alternating and symmetric groups. In a joint paper with Shareshian, the author proved that if \(G\) is an alternating or symmetric group and \(H\leq G\), then \(O_G(H)\) is not a \(D\Delta\) lattice. The present paper treats the lower signalizer lattice case proving that if \(G\) is an alternating or symmetric group, \(t>1\) and \(m_1\geq m_2\geq\cdots\geq m_t\geq 3\), then there does not exists a simple group \(L\) and \(\gamma=(H,J)\) such that \(F^*(H/J)\cong L\), \(G=\langle W_0(\gamma),H\rangle\) and \(\Xi(\gamma)\cong D\Delta(m_1,\dots,m_t)\).