19 problems

Let $\ell$ be a prime. A McKay bijection with respect to $\ell$ is a bijection between the irreducible characters of a finite group and those of an appropriate normaliser of a Sylo…

Picky+ Conjecture. The Picky+ Conjecture holds for $(G,\ell,x)$ if the Picky Conjecture holds for $(G,\ell,x)$ and, in addition, for every $\chi\in\operatorname{Irr}^x(G)$,

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 $G$, with commutator subgroup $P'=[P,P]$. Galois McKay conjecture. The exponent of the group…

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 $G$ be a finite group, let $p$ be a prime, let $P\in\operatorname{Syl}_p(G)$, and let $H=\mathbf{N}_G(P)$. Evseev's IRC-Syl conjecture. The condition…

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 $P$ be a Sylow $p$-subgroup of $G$. Write $P'=[P,P]$ for the commutator subgroup, and let $\norm GP/P$ denote the relevant se…

Let $F$ be a subfield of $\mathbb{C}$ of characteristic zero. Let $G$ be a finite group, let $P$ be a Sylow $p$-subgroup of $G$, and set $N=N_G(P)$. Let $\mathcal{P}$ and…

Let $G$ be a finite group, let $p$ be a prime, let $P$ be a Sylow $p$-subgroup of $G$, and set $N=N_G(P)$. For an integer $i$ not divisible by $p$, let $M_i(G)$ denote the number o…

Let $G$ be a finite simple group, let $P$ be a Sylow $2$-subgroup of $G$, and let $N=N_G(P)$. McKay's conjecture. The groups $G$ and $N$ have the same number of irreducible charact…

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 $N_G(P)$…

McKay's conjecture. The displayed equality holds for every finite group $G$ and every prime $p$. This is the character-counting conjecture predicted by McKay and is stated here 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 $G$ be a finite group and let $\ell$ be a prime. Write $\operatorname{Irr}_{\ell'}(G)$ for the ordinary irreducible characters $\chi\in\operatorname{Irr}(G)$ such that…

Let $G$ be a finite simple group, and let $S_2$ be a Sylow $2$-subgroup of $G$. For a finite group $H$, let $m_2(H)$ denote the number of complex irreducible characters of $H$ of o…

Let $H$ be a finite group and let $p$ be a prime. Let $N$ be the normalizer in $H$ of a Sylow $p$-subgroup, and let $\operatorname{Irr}_{p'}(X)$ denote the set of irreducible chara…

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…