Finite groups with small character degrees and large prime divisors. II

Prime Divisors of Irreducible Character Degrees

Let G be a finite group. We denote by ρ(G) the set of primes which divide some character degrees of G and by σ(G) the largest number of distinct primes which divide a single character degree of G. We show that |ρ(G)| ≤ 2σ(G) + 1 when G is an almost simple group. For arbitrary finite groups G, we show that |ρ(G)| ≤ 2σ(G) + 1 provided that σ(G) ≤ 2.

On Finite Groups With Two Irreducible Character Degrees

Nonsolvable Groups with No Prime Dividing Three Character Degrees

Throughout this note, G will be a finite group, Irr(G) will be the set of irreducible characters of G, and cd(G) will be the set of character degrees of G. We consider groups where no prime divides at least three degrees in cd(G). Benjamin studied this question for solvable groups in [1]. She proved that solvable groups with this property satisfy |cd(G)| 6 6. She also presented examples to show...

on finite groups with two irreducible character degrees

Finite groups with three relative commutativity degrees

‎‎For a finite group $G$ and a subgroup $H$ of $G$‎, ‎the relative commutativity degree of $H$ in $G$‎, ‎denoted by $d(H,G)$‎, ‎is the probability that an element of $H$ commutes with an element of $G$‎. ‎Let $mathcal{D}(G)={d(H,G):Hleq G}$ be the set of all relative commutativity degrees of subgroups of $G$‎. ‎It is shown that a finite group $G$ admits three relative commutativity degrees if a...