Brauer’s Height Zero Conjecture for principal blocks

Abstract We prove the other half of Brauer’s Height Zero Conjecture in the case of principal blocks.

On defects of characters and decomposition numbers

We propose upper bounds for the number of modular constituents of the restriction modulo [math] of a complex irreducible character of a finite group, and for its decomposition numbers, in certain cases.

Characterizing normal Sylow p-subgroups by character degrees

Abstract Suppose that G is a finite group, let p be a prime and let P ∈ Syl p ( G ) . We prove that P is normal in G if and only if all the irreducible constituents of the permutation character ( 1 P ) G have degree not divisible by p.

Nilpotent and abelian Hall subgroups in finite groups

[EN] We give a characterization of the finite groups having nilpotent or abelian Hall pi-subgroups that can easily be verified using the character table.

Representations of Finite Groups

Blocks with Equal Height Zero Degrees

We study blocks all of whose height zero ordinary characters have the same degree. We suspect that these might be the Broue-Puig nilpotent blocks.

Nonsolvable groups with few character degrees

Real characters of p′-degree

Nondivisibility among character degrees II: Nonsolvable groups

We say that a finite group G is an NDAD-group (no divisibility among degrees) if for any 1 < a < b in the set of degrees of the complex irreducible characters of G, a does not divide b. In this article, we determine the nonsolvable NDAD-groups. Together with the work of Lewis, Moreto and Wolf (J. Group Theory 8 (2005)), this settles a problem raised by Berkovich and Zhmud’, which asks for a classification of the NDAD-groups.

Self-normalizing Sylow subgroups

Using the classification of finite simple groups we prove the following statement: Let p &gt; 3 p&gt;3 be a prime, Q Q a group of automorphisms of p p -power order of a finite group G G , and P P a Q Q -invariant Sylow p p -subgroup of G G . If C N G ( P ) / P ( Q ) \mathbf {C}_{\mathbf {N}_G(P)/P}(Q) is trivial, then G G is solvable. An equivalent formulation is that if G G has a self-normalizing Sylow p p -subgroup with p &gt; 3 p &gt;3 a prime, then G G is solvable. We also investigate the possibilities when p = 3 p=3 .

Defect zero characters predicted by local structure

Let $G$ be a finite group and let $p$ be a prime. Assume that there exists a prime $q$ dividing $|G|$ which does not divide the order of any $p$-local subgroup of $G$. If $G$ is $p$-solvable or $q$ divides $p-1$, then $G$ has a $p$-block of defect zero. The case $q=2$ is a well-known result by Brauer and Fowler.

A Dual Version of Huppert's  -  Conjecture

Huppert’s ρ-σ conjecture asserts that any finite group has some character degree that is divisible by “many” primes. In this note, we consider a dual version of this problem, and we prove that for any finite group there is some prime that divides “many” character degrees.

Conjugacy class numbers and π-subgroups