A Characterization of $L_2(2^f)$ in Terms of Character Zeros

For a character $\chi$ of a finite group $G$, the number cod$(\chi):=|G:\mathrm{ker}(\chi)|/\chi(1)$ is called the codegree of $\chi$.In this paper, we give a solvability criterion for a finite group $G$ depending on the minimum of the ratio $\chi(1)^2 /\mathrm{cod}(\chi )$, when $\chi $ varies among the irreducible characters of $G$.

An element $x$ in a finite group $G$ is said to be \textit{vanishing} if some (complex) irreducible character of $G$ takes value $0$ at $x$. In this article, we prove that every finite simple group, except $\mathrm{SL}_2(4)$ and $\mathrm{SL}_2(8)$, contains a vanishing element of prime power order whose conjugacy class size is divisible by three distinct primes. We use this result to prove that if $p$ is a prime, which does not divide the conjugacy class size of any vanishing...

It is known that, if all the real-valued irreducible characters of a finite group have odd degree, then the group has normal Sylow $2$-subgroup. We generalize this result for Sylow $p$-subgroups, for any prime number $p$, while assuming the group to be $p$-solvable. In particular, it is proved that a $p$-solvable group has a normal Sylow $p$-subgroup if $p$ does not divide the degree of any irreducible character of the group fixed by a field automorphism of order $p$.

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.

Let $G$ be a finite solvable group, let $Irr(G)$ be the set of all complex irreducible characters of $G$ and let $cd(G)$ be the set of all degrees of characters in $Irr(G).$ Let $\rho(G)$ be the set of primes that divide degrees in $cd(G).$ The character degree graph $\Delta(G)$ of $G$ is the simple undirected graph with vertex set $\rho(G)$ and in which two distinct vertices $p$ and $q$ are adjacent if there exists a character degree $r \in cd(G)$ such that $r$ is divisible ...

In this paper, we study the structures of finite groups using some arithmetic conditions on the sizes of real conjugacy classes. We prove that a finite group is solvable if the prime graph on the real class sizes of the group is disconnected. Moreover, we show that if the sizes of all non-central real conjugacy classes of a finite group $G$ have the same $2$-part and the Sylow $2$-subgroup of $G$ satisfies certain condition, then $G$ is solvable.

We develop in this work a method to parametrize the set $\mathrm{Irr}(U)$ of irreducible characters of a Sylow $p$-subgroup $U$ of a finite Chevalley group $G(p^f)$ which is valid for arbitrary primes $p$, in particular when $p$ is a very bad prime for $G$. As an application, we parametrize $\mathrm{Irr}(U)$ when $G=\mathrm{F}_4(2^f)$.

We classify the finite groups $G$ which satisfies the condition that every complex irreducible character,whose degree's square doesn't divide the index of its kernel in $G$, lies in the same Galois conjugacy class.

Let $G$ be a finite group, and write ${\rm cd}(G)$ for the degree set of the complex irreducible characters of $G$. The group $G$ is said to satisfy the {\it two-prime hypothesis} if, for any distinct degrees $a, b \in {\rm cd}(G)$, the total number of (not necessarily different) primes of the greatest common divisor ${\rm gcd}(a, b)$ is at most $2$. In this paper, we prove an upper bound on the number of irreducible character degrees of a nonsolvable group that has a composi...

Let p be a prime larger than 3 and let G be a finite group. We prove that G is p-solvable of p-length at most 2 if there are at most two distinct character degrees relatively prime to p in the principal p-block of G. This generalizes a theorem of Isaacs-Smith, as well as a recent result of three of the present authors.