Related papers: Character codegrees, kernels, and Fitting heights …

Let $G$ be a finite group and $Irr(G)$ the set of irreducible complex characters of $G$. Let $e_p(G)$ be the largest integer such that $p^{e_p(G)}$ divides $\chi(1)$ for some $\chi \in Irr(G)$. We show that $|G:\mathbf{F}(G)|_p \leq p^{k…

Let $G$ be a finite solvable group with disconnected character degree graph $\Delta(G)$. Under these conditions, it follows from a result of P\'alfy that $\Delta(G)$ consists of two connected components. Another result of P\'alfy's gives an…

Let $G$ be a finite group, and let cs$(G)$ be the set of conjugacy class sizes of $G$. Recalling that an element $g$ of $G$ is called a \emph{vanishing element} if there exists an irreducible character of $G$ taking the value $0$ on $g$, we…

Let $G$ be a finite group. An element $g$ of $G$ is called a vanishing element if there exists an irreducible character $\chi$ of $G$ such that $\chi(g) = 0$; in this case, we say that the conjugacy class of $g$ is a vanishing conjugacy…

We prove that if $p$ is an odd prime, $G$ is a solvable group, and the average value of the irreducible characters of $G$ whose degrees are not divisible by $p$ is strictly less than $2(p+1)/(p+3)$, then $G$ is $p$-nilpotent. We show that…

Let $G$ be a finite group, and $\alpha$ a nontrivial character of $G$. The McKay graph $\mathcal{M}(G,\alpha)$ has the irreducible characters of $G$ as vertices, with an edge from $\chi_1$ to $\chi_2$ if $\chi_2$ is a constituent of…

For a finite group $G$, let $\Delta(G)$ denote the character graph built on the set of degrees of the irreducible complex characters of $G$. Akhlaghi and Tong-Viet in \cite{[AT]} conjectured that if for some positive integer $n$,…

We develop the concept of character level for the complex irreducible characters of finite, general or special, linear and unitary groups. We give characterizations of the level of a character in terms of its Lusztig's label and in terms of…

The classification of irreducible, spherical characters of the infinite-dimensional unitary/orthogonal/symplectic groups can be obtained by finding all possible limits of normalized, irreducible characters of the corresponding…

Motivated by the Problem $164$ proposed by Y. Berkovich and E. Zhmud' in their book "Characters of finite groups, Part $1$", we give a characterization of finite groups whose irreducible character codegrees are prime powers. This is based…

Let $G$ be a finite soluble group and $h(G)$ its Fitting length. The aim of this paper is to give certain upper bounds for $h(G)$ as functions of the Fitting length of at least three Hall subgroups of $G$ which factorize $G$ in a particular…

Let J be a finite-dimensional nilpotent algebra over a finite field F_q. We formulate a procedure for analysing characters of the group 1+J. In particular, we study characters of the group $U_n (q)$ of unipotent triangular $n\times n$…

We are interested in determining the bound of the average of the degrees of the irreducible characters whose degrees are not divisible by some prime $p$ that guarantees a finite group $G$ of odd order is $p$-nilpotent. We find a bound that…

We study the irreducibility of 6-dimensional strictly compatible systems of Q with distinct Hodge-Tate weights. We prove that if one of the representations $\rho$ in such a system is irreducible and satisfies a self-dual condition…

Let $p$ be a prime such that $p \geq 5$. Let $G$ be a finite $p$-solvable group and let $p^a$ be the largest power of $p$ dividing $\chi(1)$ for an irreducible character $\chi$ of $G$, we show that $|G:F(G)|_p \leq p^{5.5a}$. Let $G$ be a…

Let $q$ be a prime power and $U$ the group of lower unitriangular matrices of order $n$ for some natural number $n$. We give a lower bound for the degrees of irreducible constituents of Andr\'{e}-Yan supercharacters and classify the…

Let \(G\) be a finite solvable group, and let \(\Delta(G)\) denote the \emph{prime graph} built on the set of degrees of the irreducible complex characters of \(G\). A fundamental result by P.P. P\'alfy asserts that the complement…

Goldmann and Russell (2002) initiated the study of the complexity of the equation satisfiability problem in finite groups by showing that it is in P for nilpotent groups while it is NP-complete for non-solvable groups. Since then, several…

We classify the quasi-finite irreducible highest weight modules over the infinite rank Lie superalgebras $\hgltwo$, $\hC$ and $\hD$, and determine the necessary and sufficient conditions for quasi-finite irreducible highest weight modules…

For a finite non-abelian group $G$ let $\rat(G)$ denote the largest ratio of degrees of two nonlinear irreducible characters of $G$. We prove that the number of non-abelian composition factors of $G$ is bounded above by…