Mariagrazia Bianchi

Character degree graphs that are complete graphs

Let G be a finite group, and write cd(G) for the set of degrees of irreducible characters of G. We define F(G) to be the graph whose vertex set is cd(G) - {1}, and there is an edge between a and b if (a, b) > 1. We prove that if Γ(G) is a complete graph, then G is a solvable group.

Finite groups and subgroup-permutability

Subgrouppermutability in a finite group is studied, in order to distinguish between irreducible and reducible groups and to give a bound for the diameter. A simple nonabelian finite group is irreducible and its diameter is less or equal16 while a reducible group is forced to be soluble and the bound of the diameter of such a group is 4.

On the regularity of a graph related to conjugacy classes of groups

Given a finite group G , denote by ? ( G ) the simple undirected graph whose vertices are the (distinct) non-central conjugacy class sizes of G , and for which two vertices of ? ( G ) are adjacent if and only if they are not coprime numbers. In this note we prove that ? ( G ) is a 2 -regular graph if and only if it is a complete graph with three vertices, and ? ( G ) is a 3 -regular graph if and only if it is a complete graph with four vertices.

Subgroup-Permutability and Affine Planes*

We classify all finite groups whose subgroup-permutability graph is the incidence graph of an affine plain.

Maximal supersoluble subgroups of symmetric groups

SummaryAll maximal supersoluble subgroups of symmetric groups are classified.