Methods and questions in character degrees of finite groups

We present some variations on some of the main open problems on character degrees. We collect some of the methods that have proven to be very useful to work on these problems. These methods are also useful to solve certain problems on zeros of characters, character kernels and fields of values of characters

The average character degree of finite groups and Gluck's conjecture

We prove that the order of a finite group $G$ with trivial solvable radical is bounded above in terms of ${\rm acd}(G)$, the average degree of the irreducible characters. It is not true that the index of the Fitting subgroup is bounded above in terms of ${\rm acd}(G)$, but we show that in certain cases it is bounded in terms of the degrees of the irreducible characters of $G$ that lie over a linear character of the Fitting subgroup. This leads us to propose a refined version of Gluck's conjecture.Comment: Fixed an inaccuracy in the statement of Lemma 2.

pp-groups and zeros of characters

Fix a prime $p$ and an integer $n\geq 0$. Among the non-linear irreducible characters of the $p$-groups of order $p^n$, what is the minimum number of elements that take the value 0?Comment: to appear in Arch. Mat

Common zeros of irreducible characters

We study the zero-sharing behavior among irreducible characters of a finite group. For symmetric groups $S_n$, it is proved that, with one exception, any two irreducible characters have at least one common zero. To further explore this phenomenon, we introduce the common-zero graph of a finite group $G$, with non-linear irreducible characters of $G$ as vertices, and edges connecting characters that vanish on some common group element. We show that for solvable and simple groups, the number of connected components of this graph is bounded above by 3. Lastly, the result for $S_n$ is applied to prove the non-equivalence of the metrics on permutations induced from faithful irreducible characters of the group

Influence of the number of Sylow subgroups on solvability of finite groups

Let $G$ be a finite group. We prove that if the number of Sylow $3$-subgroups of $G$ is at most $7$ and the number of Sylow $5$-subgroups of $G$ is at most $1455$, then $G$ is solvable. This is a strong form of a recent conjecture of Robati

Influence of the number of Sylow subgroups on solvability of finite groups

Let $G$ be a finite group. We prove that if the number of Sylow $3$-subgroups of $G$ is at most $7$ and the number of Sylow $5$-subgroups of $G$ is at most $1455$, then $G$ is solvable. This is a strong form of a recent conjecture of Robati