For a finite group G, let cd(G) be the set of irreducible complex character degrees of G forgetting multiplicities and X1(G) be the set of all irreducible complex character degrees of G counting multiplicities. Suppose that p is a prime number. We prove that if G is a finite group such that |G| = |PGL(2, p)|, p ∈ cd(G) and max(cd(G)) = p+1, then G ∼= PGL(2, p), SL(2, p) or PSL(2, p) × A, where ...

1. Let k be an algebraic closure of a finite field Fq. Let G = GLn(k). The group G(Fq) = GLn(Fq) can be regarded as the fixed point set of the Frobenius map F : G −→ G, (gij) 7→ (g q ij). Let Q̄l be an algebraic closure of the field of l-adic numbers, where l is a prime number invertible in k. The characters of irreducible representations of G(Fq) over an algebraically closed field of characteri...

In a previous paper, the second author established that, given finite fields F < E and certain subgroups C ≤ E×, there is a Galois connection between the intermediate field lattice {L | F ≤ L ≤ E} and C’s subgroup lattice. Based on the Galois connection, the paper then calculated the irreducible, complex character degrees of the semi-direct product C⋊Gal(E/F ). However, the analysis when |F | i...

There are some variations of theorems A and B which are simply not true. For instance, if χ in Irr(G) has degree divisible by p, then there does not necessarily exist a p-element on which χ vanishes. It is enough to consider L2(11) with any character of degree 10 and p = 2. It is also not true that if χ vanishes on some element x, then χ has to vanish on some p-part of x. For instance, if G is ...

Abstract Let G be a finite group. Denoting by $$\textrm{cd}(G)$$ cd ( G ) the set of degrees irreducible complex characters , we consider character degree graph : this is (simple undirected) whose vertices are prime divisors numbe...

Let H denote a semisimple Hopf algebra over an algebraically closed field k of characteristic 0. We show that the degree of any irreducible representation of H whose character belongs to the center of H∗ must be a divisor of dimk H .

We exhibit for each integer n ≥ 15 an ordinary irreducible character of the symmetric group Sn, which restricts irreducibly to An, with the property that its degree is divisible by every prime less than or equal to n, thereby proving a conjecture of D. L. Alvis.

Isaacs has defined a character to be super monomial if every primitive character inducing it is linear. Isaacs has conjectured that if G is an M -group with odd order, then every irreducible character is super monomial. We prove that the conjecture is true if G is an M -group of odd order where every irreducible character is a {p}-lift for some prime p. We say that a group where irreducible cha...