Speaker: Eda Kaja (Darmstadt)
Title: Classification of Non-Solvable Groups whose Power Graph is a Cograph
A recent, active branch of research in algebraic graph theory studies constructions of graphs whose vertex set is a group \(G\) and whose edges reflect the structure of \(G\) in some way. An important example of such a graph is the power graph of a group \(G\). Its vertices are the elements of \(G\) and there is an edge between distinct vertices \(x\) and \(y\) of \(G\) if and only if \(x\) is a power of \(y\) or \(y\) is a power of \(x\).
We are interested in groups whose power graph is a cograph, i.e. it does not contain an induced subgraph isomorphic to a path of length four. We call such groups power-cograph groups. The problem of classifying power-cograph groups was posed by Cameron, Manna and Mehatari. They solved this problem for nilpotent groups and provided a classification in the case of finite simple groups relative to number theoretic problems. In this talk, I will present our classification of non-solvable power-cograph groups relative to the same number theoretic problems. Additionally, our techniques allow us to precisely describe the structure of solvable power-cograph groups.
This is joint work with Jendrik Brachter.