Fitting heights of solvable groups with no nontrivial prime power 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

AFFINE SUBGROUPS OF THE CLASSICAL GROUPS AND THEIR CHARACTER DEGREES

In this paper we describe how the degrees of the irreducible characters of the affine subgroups of the classical groups under consideration can be found inductively. In [4] Gow obtained certain character degrees for all of the affine subgroups of the classical groups. We apply the method of Fischer to the above groups and, in addition to the character degrees given in [4], we obtain some ne...

Finite groups with $X$-quasipermutable subgroups of prime power order

Let $H$, $L$ and $X$ be subgroups of a finite group$G$. Then $H$ is said to be $X$-permutable with $L$ if for some$xin X$ we have $AL^{x}=L^{x}A$. We say that $H$ is emph{$X$-quasipermutable } (emph{$X_{S}$-quasipermutable}, respectively) in $G$ provided $G$ has a subgroup$B$ such that $G=N_{G}(H)B$ and $H$ $X$-permutes with $B$ and with all subgroups (with all Sylowsubgroups, respectively) $...

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.