9 problems

Picky Conjecture. There exists a bijection

Let $G$ be a finite group, let $p$ be a prime, let $P\in\operatorname{Syl}_p(G)$, let $H=\mathbf{N}_G(P)$, and let $\mathcal P$ be the set of picky elements of $P$. Define…

Let $S$ be a quasisimple group with cyclic center, let $p$ be a prime, let $P\in\operatorname{Syl}_p(S)$, and set $A=\operatorname{Aut}(S)_P$. For…

Let $S$ be a quasisimple group with cyclic center, let $p$ be a prime, let $P\in\operatorname{Syl}_p(S)$, and set $A=\operatorname{Aut}(S)_P$. For…

Let $G$ be a finite group, let $p$ be a prime, and let $P\in\operatorname{Syl}_p(G)$. Write $\operatorname{Irr}_{p'}(G)$ for the irreducible complex characters of $G$ whose degrees…

Let $G$ be a finite group, let $p$ be a prime, and let $P$ be a Sylow $p$-subgroup of $G$. Let $B_0$ be the principal $p$-block of $G$ and $b_0$ the principal $p$-block of $N_G(P)$…

Navarro's refinement. There exists a bijection between the $\sigma$-invariant characters in

2-McKay conjecture. For each character $\chi$, the 2-representations $\Theta_H(\chi)$ and $\Theta_N(\chi)$ of $G$ are isomorphic. This is a 2-representation-theoretic strengthening…

Let $(\mathbf G,F)$ be a classical group with regular number $d$, and let a very good Sylow $d$-twist mean a Sylow $d$-twist having the maximal extensibility property described abo…