Related papers: Character degrees in principal blocks for distinct…

An irreducible ordinary character of a finite reductive group is called quadratic unipotent if it corresponds under Jordan decomposition to a semisimple element $s$ in a dual group such that $s^2=1$. We prove that there is a bijection…

If a finite group $A$ acts coprimely as automorphisms on a finite group $G$, then the $A$-invariant Brauer $p$-blocks of $G$ are exactly those that contain $A$-invariant irreducible characters.

Milliet asks the following question: given two prime numbers $p\neq q$, is there a division algebra of characteristic $p$ which is of dp-rank $q^2$ and of dimension $q^2$ over its center? We answer in the affirmative. We also give an…

We show that the largest character degree of an alternating group $A_n$ with $n\geq 5$ can be bounded in terms of smaller degrees in the sense that \[ b(A_n)^2<\sum_{\psi\in\textrm{Irr}(A_n),\,\psi(1)< b(A_n)}\psi(1)^2, \] where…

We show that a group is a GVZ-group if and only if it is a flat group. We show that the nilpotence class of a GVZ-group is bounded by the number of distinct degrees of irreducible characters. We also show that certain CM-groups can be…

We prove, for primes $p\ge5$, two inequalities between the fundamental invariants of Brauer $p$-blocks of finite quasi-simple groups: the number of characters in the block, the number of modular characters, the number of height zero…

Let \chi be an irreducible character of the finite group G. If g is an element of G and \chi(g) is not zero, then we conjecture that the order of g divides |G|/\chi(1). The conjecture is a generalization of the classical fact that…

Let $G$ be a finite group and $p\in \pi(G)$, and let Irr$(G)$ be the set of all irreducible complex characters of $G$. Let $\chi \in {\rm Irr}(G)$, we write ${\rm cod}(\chi)=|G:{\rm ker} \chi|/\chi(1)$, and called it the codegree of the…

Let $N$ be a normal subgroup of a finite group $G$. In this paper, we consider the elements $g$ of $N$ such that $\chi(g)\neq 0$ for all irreducible characters $\chi$ of $G$. Such an element is said to be non-vanishing in $G$. Let $p$ be a…

Let $G$ be a finite group of order divisible by a prime $p$. The number of $p$-regular and $p'$-regular conjugacy classes of $G$ is at least $2\sqrt{p-1}$. Also, the number of $p$-rational and $p'$-rational irreducible characters of $G$ is…

Let $q$ be an odd prime power, $n > 1$, and let $P$ denote a maximal parabolic subgroup of $GL_n(q)$ with Levi subgroup $GL_{n-1}(q) \times GL_1(q)$. We restrict the odd-degree irreducible characters of $GL_n(q)$ to $P$ to discover a…

A classical theorem on character degrees states that if a finite group has fewer than four character degrees, then the group is solvable. We prove a corresponding result on character values by showing that if a finite group has fewer than…

The prime graph of a finite group $G$, which is denoted by ${\rm GK}(G)$, is a simple graph whose vertex set is comprised of the prime divisors of $|G|$ and two distinct prime divisors $p$ and $q$ are joined by an edge if and only if there…

Let $G$ be a finite group and $\mathrm{Irr}(G)$ the set of all irreducible complex characters of $G$. Define the codegree of $\chi \in \mathrm{Irr}(G)$ as $\mathrm{cod}(\chi):=\frac{|G:\mathrm{ker}(\chi) |}{\chi(1)}$ and denote by…

Let $G$ be a finite solvable group, let $p$ be a prime such that $p \geq 5$ and $O_p(G)=1$, and we denote $|G|_p=p^n$, then $G$ contains a block of defect less than or equal to $\lfloor \frac {3n} 5 \rfloor$. Let $G$ be a finite solvable…

A character of a finite group having degree $n$ takes values which may be expressed as sums of $n$ or fewer roots of unity. In this note, we prove a result which describes the irreducible constituents of generalized characters on abelian…

Let $G$ be a finite group, and let $\Delta(G)$ be the prime graph built on its set of conjugacy class sizes: this is the (simple undirected) graph whose vertices are the prime numbers dividing some conjugacy class size of $G$, and two…

Let $G$ be a finite group and $d$ the degree of a complex irreducible character of $G$, then write $|G|=d(d+e)$ where $e$ is a nonnegative integer. We prove that $|G|\leq e^4-e^3$ whenever $e>1$. This bound is best possible and improves on…

The classical It\^{o}-Michler theorem states that the degree of every ordinary irreducible character of a finite group $G$ is coprime to a prime $p$ if and only if the Sylow $p$-subgroups of $G$ are abelian and normal. In an earlier paper,…

Let $S_n$ denote a symmetric group, $\chi$ an irreducible character of $S_n$, and $g\in S_n$ an element which decomposes into $k$ disjoint cycles, where $1$-cycles are included. Then $|\chi(g)|\le k!$, and this upper bound is sharp for…