3 problems

Let $G$ be a finite group, let $p$ be a prime, and let $P\in\operatorname{Syl}_p(G)$. Define … and let $Q\in\operatorname{Syl}_p(K)$. Call a character almost $p$-rational when it h…

Let $G$ be a finite group and $p$ a prime. Let $B_0(G)$ denote the principal $p$-block of $G$, and write $c(\theta)_p=p^r$ for the $p$-rationality level of a character $\theta$. Hu…

Navarro–Tiep conjecture. One has $d_{\chi 1_G}\neq 0$ for every $\chi\in{\operatorname{Irr}}(B_0(G))$ if and only if $G$ has a normal $p$-complement.