On the diameter of a graph related to \(p\)-regular conjugacy classes of finite groups
Fix a prime \(p\) and a finite group \(G\). Consider the conjugacy classes of \(G\) for elements of order prime to \(p\), call this set \(\text{Con}_p(G)\). Now mimic a well-known construction of a graph by choosing as vertices the primes which divide the sizes of the classes in \(\text{Con}_p(G)\). Join two vertices if there is an element in \(\text{Con}_p(G)\) which is divisible by both primes. The authors prove a number of theorems about this graph for \(p\)-soluble groups. For example that the number of connected components is at most 2 (Theorem 1). If \(G\) is soluble with two connected components then the primes dividing \(|G|\) other than \(p\) can be divided into two sets, \(\pi_1\) and \(\pi_2\) so that at least one Hall \(\pi_i\)-subgroup is metabelian.
scientific article; zbMATH DE number 1526433