Irreducible characters taking root of unity values on $p$-singular elements

In this paper we study finite p-solvable groups having irreducible complex characters chi in Irr(G) which take roots of unity values on the p-singular elements of G.

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.

On Real and Rational Characters in Blocks

Abstract The principal $p$-block of a finite group $G$ contains only one real-valued irreducible ordinary character exactly when $G/{{\bf O}_{p'}(G)}$ has odd order. For $p \ne 3$, the same happens with rational-valued characters. We also prove an analogue for $p$-Brauer characters with $p \geq 3$.

Blocks with 𝑝-power character degrees

Let B B be a p p -block of a finite group G G . If χ ( 1 ) \chi (1) is a p p -power for all χ ∈ Irr ⁡ ( B ) \chi \in \operatorname {Irr}(B) , then B B is nilpotent.

Conjugacy class numbers and π-subgroups