Related papers: Feit numbers and $p'$-degree characters

A classical theorem of I. Schur states that the degree of any irreducible complex representation of a finite group $G$ divides the order of $G/\mathscr{Z} G$, where $\mathscr{Z} G$ is the center $G$. This note discusses similar divisibility…

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…

We shall give an explicit upper bound for the smallest prime factor of multiperfect numbers of the form $N=p_1^{\alpha_1}\cdots p_s^{\alpha_s} q_1^{\beta_1}\cdots q_t^{\beta_t}$ with $\beta_1, \ldots, \beta_t$ bounded by a given constant.…

Let G be a finite group of order n and V an irreducible representation over the complex numbers of dimension d. For some nonnegative number e, we have n=d(d+e). If e is small, then the character of V has unusually large degree. We fix e and…

Let $p$ be a prime, $m$ be a positive integer ( $m \geq 1$, and $m \geq 2$ if $p=2$), and $\chi_n$ be a multiplicative complex character on $\mathbb F^*_{p^m}$ with order $n| (p^m-1)$. We show that a partition $\mathcal A_1 \cup \mathcal…

A complex irreducible character of a finite group G is said to be p-constant, for some prime p dividing the order of G, if it takes constant value at the set of p-singular elements of G. In this paper we classify irreducible p-constant…

In this paper, we show that if $p$ is a prime and $G$ is a $p$-solvable group, then $| G:O_p (G) |_p \le (b(G)^p/p)^{1/(p-1)}$ where $b(G)$ is the largest character degree of $G$. If $p$ is an odd prime that is not a Mersenne prime or if…

We prove a variation of Thompson's Theorem. Namely, if the first column of the character table of a finite group $G$ contains only two distinct values not divisible by a given prime number $p>3$, then $O^{pp'pp'}(G)=1$. This is done by…

Let $G$ be a finite group and ${\rm cd}(G)$ will be the set of the degrees of the complex irreducible characters of $G$. Also let ${\rm cod}(G)$ be the set of codegrees of the irreducible characters of $G$. The Taketa problem conjectures if…

We prove that there exists an integer-valued function f on positive integers such that if a finite group G has at most k real-valued irreducible characters, then |G/Sol(G)| is at most f(k), where Sol(G) denotes the largest solvable normal…

In this paper, we get the sharp bound for $|G/O_p(G)|_p$ under the assumption that either $p^2 \nmid \chi(1)$ for all $\chi \in {\rm Irr}(G)$ or $p^2 \nmid \phi(1)$ for all $\phi \in {\rm IBr}_p(G)$. This would settle two conjectures raised…

Let $G$ be a finite group and ${\rm cd}(G)$ denote the set of complex irreducible character degrees of $G$. In this paper, we prove that if $G$ is a finite group and $H$ is an almost simple group with socle $H_{0}= \, ^{2}{\rm G}_{2}(q)$,…

Let $q$ be a power of a prime $p$ and let $U(q)$ be a Sylow $p$-subgroup of a finite Chevalley group $G(q)$ defined over the field with $q$ elements. We first give a parametrization of the set $\text{Irr}(U(q))$ of irreducible characters of…

Most integers are composite and most univariate polynomials over a finite field are reducible. The Prime Number Theorem and a classical result of Gau{\ss} count the remaining ones, approximately and exactly. For polynomials in two or more…

We prove a variant of the Theorem of Ito-Michler, investigating the properties of finite groups where a prime number $p$ does not divide the degree of any irreducible character left invariant by some Galois automorphism $\sigma$ of order…

In this note, we prove that if $G$ is solvable and ${\rm cod}(\chi)$ is a $p$-power for every nonlinear, monomial, monolithic $\chi\in {\rm Irr}(G)$ or every nonlinear, monomial, monolithic $\chi \in {\rm IBr} (G)$, then $P$ is normal in…

For an irreducible character $\chi$ of a finite group $G$, let $\mathrm{cod}(\chi):=|G: \ker(\chi)|/\chi(1)$ denote the codegree of $\chi$, and let $\mathrm{cod}(G)$ be the set of irreducible character codegrees of $G$. In this note, we…

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 X be an irreducible, primitive complex character of the finite solvable group G, and let X* denote the complex conjugate character. If the degree X(1) is odd, then we show how to associate to X in a unique way, a conjugacy class of…

Let G be a finite group and ? be an irreducible character of G, the number cod(?) = jG : Let $ G $ be a finite group and $ \chi $ be an irreducible character of $ G $, the number $ \cod(\chi) = |G: \kernel(\chi)|/\chi(1) $ is called the…