Nonsolvable Groups with No Prime Dividing Three Character Degrees

Nonsolvable Groups All of Whose Character Degrees Are Odd-Square-Free

A finite group G is odd-square-free if no irreducible complex character of G has degree divisible by the square of an odd prime. We determine all odd-square-free groups G satisfying S 6 G 6 Aut(S) for a finite simple group S. More generally, we show that ifG is any nonsolvable odd-square-free group, then G has at most two nonabelian chief factors and these must be simple odd-square-free groups....

Four-Vertex Degree Graphs of Nonsolvable Groups

For a finite group G, the character degree graph ∆(G) is the graph whose vertices are the primes dividing the degrees of the ordinary irreducible characters of G, with distinct primes p and q joined by an edge if pq divides some character degree of G. We determine all graphs with four vertices that occur as ∆(G) for some nonsolvable group G. Along with previously known results on character degr...

Nonsolvable Groups Satisfying the One-Prime Hypothesis

Throughout this paper, G is a finite group and Irr(G) is the set of irreducible characters of G. We are particularly interested in the values these characters take on the identity of G. If χ ∈ Irr(G), then χ(1) is the degree of χ. The set of all degrees for G is written cd(G) = {χ(1) |χ ∈ Irr(G)}. In recent years, there has been much interest in finding connections between the structure of a fi...

Nonsolvable groups with few character degrees

Groups with Two Extreme Character Degrees and their Minimal Faithful Representations

for a finite group G, we denote by p(G) the minimal degree of faithful permutation representations of G, and denote by c(G), the minimal degree of faithful representation of G by quasi-permutation matrices over the complex field C. In this paper we will assume that, G is a p-group of exponent p and class 2, where p is prime and cd(G) = {1, |G : Z(G)|^1/2}. Then we will s...