We prove that if the average of the degrees of the irreducible characters of a finite group G is less than 16/5, then G is solvable. This solves a conjecture of I. M. Isaacs, M. Loukaki and the first author. We discuss related questions.

For a finite group G, the character degree graph ∆(G) is the graph whose vertices are the primes dividing the degrees of the ordinary irreducible characters of G, with distinct primes p and q joined by an edge if pq divides some character degree of G. We determine all graphs with four vertices that occur as ∆(G) for some nonsolvable group G. Along with previously known results on character degr...

Let G be a finite group, and let $$\textrm{cd}(G)$$ denote the set of degrees irreducible complex characters G. Define then character degree graph $$\Delta (G)$$ as (simple undirected) whose vertices are prime divisors numbers in , two distinct p, q adjacent if only pq divides some number . This paper continues work, started [8], toward classification non-solvable groups possesses cut-vertex, i...

In this article, we continue the classification work done in first paper of same name. With careful modifications our previous approach, are able to deduce (with two notable exceptions) which members previously introduced graph family manifest as prime character degree some solvable group.

Abstract Let G be a finite group. Denoting by $$\textrm{cd}(G)$$ cd ( G ) the set of degrees irreducible complex characters , we consider character degree graph : this is (simple undirected) whose vertices are prime divisors numbe...