328 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…

Let $G$ be a finite group and let $B$ be a $p$-block of $G$ with defect group $D$. Brauer's $k(B)$-conjecture. The number of irreducible characters in $B$ does not exceed the order…

Let $G$ be a finite group. Write … and … Here $\operatorname{Irr}(G)$ is the set of complex irreducible characters of $G$, and $\pi(n)$ is the set of prime divisors of the positive…

Let $psi_k$ denote the staircase (2-defect zero) irreducible characters under discussion, and let a character have the tensor-square property when every irreducible character occur…

Miller's conjecture. One has

Feit's conjecture. If $\chi\in\operatorname{Irr}(G)$, then $G$ contains an element whose order is $c(\chi)$.

Let $B$ be a $p$-block of a finite group $G$, with defect group $D$. For $\chi\in\operatorname{Irr}(B)$, define its height by $\chi(1)_p=|G:D|_p p^{h(\chi)}$. Let $mh(B)$ be the sm…

For a finite group $G$, let $\operatorname{Irr}(G)$ be the set of irreducible characters and define the set of irreducible character degrees by … Let $\operatorname{dl}(G)$ denote…

Let $G$ be a finite group. An $M$-group is a group whose irreducible complex characters are all monomial, meaning induced from linear characters of subgroups. A character is super-…

Let $\chi=\chi^{\phi,\lambda}$ be a primary irreducible character of $\mathrm{GU}_n(q)$, where $\phi$ is an orbit other than $\{\mathbf{1}\}$ and $\mathbf{1}$ is the trivial charac…

Let $p$ be an odd prime, let $G$ be a finite group, and let $\rho\in\operatorname{Irr}_{p'}(G)$ have conductor $p^a m$, where $a\in\mathbb{Z}_{\geq 0}$ and…

Let $G$ be a finite group and let $uell$ denote the $ell$-adic valuation. For $chiinoperatorname{Irr}(G)$, define its $ell$-defect by $dell(chi):=nuell(|G|)-nuell(chi(1))$. For a…

Harada's conjecture II. For every finite group $G$, $h(G)\in\mathbb{Z}$.

Isaacs–Navarro–Wolf's conjecture. If $G$ is solvable, then

Let $p$ be a prime, let $G$ be a finite group, let $P\in\operatorname{Syl}_p(G)$, and let $\alpha\in\mathbb{Z}^{+}$. The $p$-rationality level of an irreducible character is determ…

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…

Navarro's conjecture.

Normal Sylow subgroup conjecture. $G$ has a normal Sylow $q$-subgroup if and only if, for any $p\neq q$, $q$ does not divide the degree of any $p$-height zero irreducible character…

Let $G$ be a finite group, let $\operatorname{Irr}(G)$ denote the set of irreducible characters of $G$, let $\chi$ be an irreducible character of $G$, and let $\rho_{\mathrm{reg}}$…

Let $G\unlhd A$ be finite groups, let $p$ be a prime, and suppose that ${\mathbf{O}}_p(G)$, the largest normal $p$-subgroup of $G$, is contained in the centre of $G$. For a $p$-blo…

Malle–Navarro conjecture. The block $B$ is nilpotent if and only if all height-zero characters in $B$ have the same degree.

Primitive character divisibility conjecture. The character degree $\chi(1)$ divides the size of some conjugacy class:

Shalev's conjecture. The commutator map should yield an almost uniform distribution on $G$, more precisely,

Commutator abelianness conjecture. The commutator subgroup

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…