15 problems

Malle–Robinson conjecture. One has

Let $p$ be a prime, let $G$ be a finite $p$-solvable group, and let $\varphi\in\operatorname{IBr}_p(G)$ be an irreducible Brauer character with vertex $Q$. Let $L_\varphi$ denote t…

Let $G$ be a finite group and let $p$ be a prime. Write $|G|_{p'}$ for the prime-to-$p$ part of the order of $G$, and let $\operatorname{IBr}(G)$ denote the irreducible $p$-Brauer…

Let $G$ be a finite group and let $p$ be a prime dividing $|G|$. Write $|G|_{p'}$ for the largest divisor of $|G|$ not divisible by $p$, let $l(G)$ be the number of $p'$-conjugacy…

Let $G$ be a finite group, let $p$ be a prime dividing $|G|$, and let $B$ be a $p$-block of $G$ with defect group $D$. Let $\operatorname{Irr}_p(B)$ be the irreducible Brauer chara…

Holm–Willems local conjecture. One has

Let $p>2$, let $G$ be a $p$-solvable group, and let $\varphi\in\operatorname{IBr}(G)$ have vertex $Q$. For a vertex pair $(Q,\delta)$, let $L_\varphi(Q,\delta)$ denote the set of l…

Prime-graph complement characterization. For every strictly $T$-solvable group $G$, there exists a set $Y\subseteq\bigcup_{p\in\pi(T)}\operatorname{IBr}_p(T)$ such that

Let $G$ be a finite group of Lie type, let … -blocks, and let $\phi\in\mathcal{E}_{,s}$ be a Brauer character. Write $\mathcal{E}_s$ for the ordinary Lusztig series associated with…

Let $G$ be a finite group and $p$ a prime. Let $O_p(G)$ be the largest normal $p$-subgroup of $G$, let $\operatorname{IBr}_p(G)$ be the set of irreducible $p$-Brauer characters of…

Let $G$ be a finite quasi-simple group with centre $Z(G)$ of $p'$-order. Let $\operatorname{IBr}(G\mid\theta)$ denote the irreducible $p$-Brauer characters of $G$ lying over…

Counting conjecture. Suppose that $p$ is good for $\mathbf{G}$. Then $\alpha_\ell$ is the number of unipotent irreducible Brauer characters of $G$.

Let $G$ be a finite group, let $p$ be a prime, and let $\chi\in\operatorname{Irr}(G)$. Let $\chi^0$ be the restriction of $\chi$ to the elements of $G$ of order prime to $p$, and l…

The Brauer degree-square conjecture. One always has

The norm inequality. The inequality