13 problems

Let $G$ be a finite group, let $p$ be a prime, and let $x\in G$ be a picky $p$-element, meaning that $x$ lies in a unique Sylow $p$-subgroup. Let $P\in\operatorname{Syl}_p(G)$ cont…

Let $G$ be a finite group, let $p$ be a prime, and let $x\in G$ be a $p$-element. Write $\operatorname{Irr}^x(G)$ for the irreducible complex characters of $G$ that are nonzero at…

Let $G$ be a finite group, let $\pi$ be a set of primes, and let $H$ be a nilpotent Hall $\pi$-subgroup of $G$. Let $\mathcal H$ be the set of elements of $H$ that are $H$-picky, m…

Let $G$ be a finite group and let $x\in G$. Write $x=\prod_p x_p$ for the decomposition of $x$ into its $p$-parts. Assume that, for every prime $p$ dividing $o(x)$, the pair…

Let $G$ be a finite group with abelian Sylow $p$-subgroups. Let $P\in\operatorname{Syl}_p(G)$ and let $x\in P$ be picky. Let $S_p(x)$ be a set of representatives for the conjugacy…

Let $G$ be a finite group, let $p$ be a prime, let $P\in\operatorname{Syl}_p(G)$, and let $x\in P$. Let $R_p(x)$ be a set of representatives for the conjugacy classes in the reduce…

McKay's conjecture. There is a bijection

Weak e-local counting conjecture. For every $\ell$-block $B$ of ${\mathbf{G}}^F$,

The e-local counting conjecture. For every $\ell$-block $B$ of ${\mathbf{G}}^F$ and $d\geq 0$,

McKay's conjecture. There is a bijection between the irreducible $p'$-characters of $G$ and the irreducible $p'$-characters of $N_G(R)$.

Let $G$ be a finite group and let $b5$ be a prime dividing $|G|$. Let $P\in\operatorname{Syl}_{b5}(G)$. Write $\operatorname{Irr}_{b5'}(G)$ for the irreducible characters of $G$ wh…

Let $G$ be a finite group, let $L\unlhd G$, let $P/L\in\operatorname{Syl}_p(G/L)$, and let $\nu\in\operatorname{Irr}(L)$ be $P$-invariant. Write $\operatorname{Irr}_{p'}(G\mid\nu)$…

Let $G$ be a finite group, let $p$ be a prime, and let $P\in\operatorname{Syl}_p(G)$. Isaacs--Navarro refinement. In the situation of McKay's conjecture, there exists a bijection ……