Groups with few $p'$-character degrees

Abstract We prove a variation of Thompson's Theorem. Namely, if the first column of the character table of a finite group G contains only two distinct values not divisible by a given prime number p > 3 , then O p p ′ p p ′ ( G ) = 1 . This is done by using the classification of finite simple groups.

Nilpotent and perfect groups with the same set of character degrees

We find a pair of finite groups, one nilpotent and the other perfect, with the same set of character degrees.

A Brauer-Wielandt formula (with an application to character tables)

If a p p -group P P acts coprimely on a finite group G G , we give a Brauer-Wielandt formula to count the number of fixed points | C G ( P ) | | \textbf {C}_{G}(P) | of P P in G G . This serves to determine the number of Sylow p p -subgroups of certain finite groups from their character tables.

Characters and generation of Sylow 2-subgroups

p-Blocks relative to a character of a normal subgroup

Abstract Let G be a finite group, let N ◃ G , and let θ ∈ Irr ( N ) be a G-invariant character. We fix a prime p, and we introduce a canonical partition of Irr ( G | θ ) relative to p. We call each member B θ of this partition a θ-block, and to each θ-block B θ we naturally associate a conjugacy class of p-subgroups of G / N , which we call the θ-defect groups of B θ . If N is trivial, then the θ-blocks are the Brauer p-blocks. Using θ-blocks, we can unify the Gluck–Wolf–Navarro–Tiep theorem and Brauer's Height Zero conjecture in a single statement, which, after work of B. Sambale, turns out to be equivalent to the Height Zero conjecture. We also prove that the k ( B ) -conjecture is true i…