Character degrees and local subgroups of 𝜋-separable groups

Let G G be a finite { p , q } \{p,q \} -solvable group for different primes p p and q q . Let P ∈ Syl p ( G ) P \in \text {Syl}_{p}(G) and Q ∈ Syl q ( G ) Q \in \text {Syl}_{q}(G) be such that P Q = Q P PQ=QP . We prove that every χ ∈ Irr ( G ) \chi \in \text {Irr}(G) of p ′ p^{\prime } -degree has q ′ q^{\prime } -degree if and only if N G ( P ) ⊆ N G ( Q ) \mathbf {N}_{G}(P) \subseteq \mathbf {N}_{G}(Q) and C Q ′ ( P ) = 1 \mathbf {C}_{Q^{\prime }}(P)=1 .

Orbit sizes, character degrees and Sylow subgroups

p-Parts of character degrees and the index of the Fitting subgroup

Abstract In a solvable group G, if p 2 does not divide χ ( 1 ) for all χ ∈ Irr ( G ) , then we prove that | G : F ( G ) | p ≤ p 2 . This bound is best possible.

VARIATIONS ON THOMPSON'S CHARACTER DEGREE THEOREM

If P is a Sylow- p -subgroup of a finite p -solvable group G , we prove that G^\prime \cap \bf{N}_G(P) \subseteq {P} if and only if p divides the degree of every irreducible non-linear p -Brauer character of G. More generally if π is a set of primes containing p and G is π-separable, we give necessary and sufficient group theoretic conditions for the degree of every irreducible non-linear p -Brauer character to be divisible by some prime in π. This can also be applied to degrees of ordinary characters.

Finite Group Elements where No Irreducible Character Vanishes

AbstractIn this paper, we consider elements x of a finite group G with the property that χ(x)≠0 for all irreducible characters χ of G. If G is solvable and x has odd order, we show that x must lie in the Fitting subgroup F(G).

Erratum to “Orbit sizes, character degrees and Sylow subgroups” [Adv. Math. 184 (2004) 18–36]