Speaker: Luca Di Gravina (Halle)
Title: Closed Neighbourhoods in Maximal Intersection Graphs
Abstract:
Given a finite group \(G\) and a proper subgroup \(H\) of \(G\), the maximal intersection graph associated with the interval \([H,G]\) in the subgroup lattice of \(G\) is the graph \(\varGamma=(V,E)\), whose set of vertices \(V\) is the set of maximal subgroups of \(G\) that contain \(H\), and where two distinct vertices \(M_1\) and \(M_2\) are joined by an edge \(\{M_1,M_2\}\in E\) whenever \(M_1\cap M_2\neq H\).
In this talk, we examine how the topological information provided by the closed neighbourhoods of \(\varGamma\) can offer insight into the relationship between the graph and the structure of the interval \([H,G]\) in the subgroup lattice of \(G\).
This perspective is motivated by the role of closure operators in the study of the Möbius function of groups.