171 problems

Let $\lambda \vdash n$ be a partition. Write $\operatorname{Ev}(\lambda)$ for the set of partitions obtained from $\lambda$ in the construction used in the paper, let…

Let $G$ be a finite simple group of Lie type, let $p$ be a prime different from the defining characteristic of $G$, and let $m$ be a positive integer. For elements $g,g'\in G$, wri…

For each partition $\lambda$, let $t_\lambda$ denote a formal variable corresponding to the normalized character value $\chi(\lambda)/\chi(1)$, and let $T_\lambda$ be the specified ratio…

For every finite group $G$, every prime $p$, and every Sylow $p$-subgroup $P\in\operatorname{Syl}_p(G)$, let $\operatorname{Irr}_p(G)=\{\chi\in\operatorname{Irr}(G):p\mid\chi(1)\}$…

For every finite solvable group $G$, every element $g\in G$ such that $\chi(g)\neq 0$ for every irreducible complex character $\chi\in\operatorname{Irr}(G)$ belongs to the Fitting…

For $n\geq 0$, let $\Sigma_{k,2n}$ denote the component of weight $k+1-2n$ in the Kerov character polynomial $\Sigma_k$, and write $[R_2^i]\Sigma_{k,2n}$ for the coefficient of…

Alperin's Conjecture C. If $P$ is TI, then

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…

Let $G$ be a finite group, and let $H$ be a finite non-abelian simple group. Denote by the character degrees of a group the degrees of its complex irreducible characters. Huppert's…

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 number, and let $P$ be a Sylow $p$-subgroup of $G$. Write $N_G(P)$ for the normalizer of $P$ in $G$, and let…

Let $G$ be a finite group and let $u_\ell$ denote the $\ell$-adic valuation. For $\chi\in\operatorname{Irr}(G)$, define its $\ell$-defect by…

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 and let $n$ be a positive integer. An irreducible character of $G$ has conductor $n$ when its character field has conductor $n$. Feit's conjecture. If $G$…

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…

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 $p$ 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…

Picky Conjecture. There exists a bijection

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 and let $\chi\in\operatorname{Irr}(G)$. Here $o(g)$ denotes the order of $g$, and $|G|/\chi(1)$ is the codegree of $\chi$. Wilde's conjecture. If…

Let $G$ be a finite group and let $H$ be a subgroup. Define the subnormalizer … and the subnormalizer subgroup $\operatorname{Sub}_G(H)=\langle S_G(H)\rangle$. For a cyclic subgrou…

Moretó–Rizo's character correspondence conjecture. For any $p$-element $x\in G$, there exists a bijection

Let $p$ be a prime, let $G$ be a $p$-solvable group, and let $\varphi\in\operatorname{IBr}(G)$ be an irreducible Brauer character. A lift of $\varphi$ is a character…

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…