148 problems

Let $p$ be a prime, let $(\mathcal{K},\mathcal{O},k)$ be a complete $p$-modular system with algebraically closed residue field $k$ of characteristic $p$, and let $G$ be a finite gr…

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…

Donovan's conjecture. For a fixed defect group, there are only finitely many block algebras of group-block pairs up to Morita equivalence.

Let $F$ be a field of characteristic $p$, and let two Specht modules for a symmetric group correspond to partitions. The $p$-core of a partition is obtained by removing rim $p$-hoo…

Weak Donovan conjecture. There exists a constant $c(D)\in\mathbb N$ such that, if $G$ is a finite group and $B$ is a block of $kG$ with defect group isomorphic to $D$, then the ent…

Assume that $k$ is an algebraically closed field, and let $P$ be a fixed finite $p$-group. A source algebra of a block is regarded as an interior $P$-algebra when the block has def…

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…

Let $\mathcal{O}$ be an arbitrary complete discrete valuation ring, let $G$ be a finite group, and let $b$ be a block idempotent of $G$ over $\mathcal{O}$ with an abelian defect gr…

Malle–Robinson conjecture. One has

Blockwise Alperin weight conjecture. For every $p$-block $B$ of every finite group,

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 $G\unlhd A$ 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.

Robinson's conjecture. One has

Bounded Morita Frobenius number conjecture. Amongst all finite groups $G$ and blocks $B$ of $kG$ with defect group isomorphic to $D$, the Morita Frobenius number…

Let $G$ be a finite group, let $p$ be a prime, and let $B$ be a Brauer $p$-block of $G$ with defect group $D$. Write $D'=[D,D]$ for the derived subgroup of $D$. Let $k(B)$ be the n…

Let $B$ be a $3$-block with metacyclic, minimal non-abelian defect group $D$, let $\mathcal{F}$ be its associated fusion system, and let $k_i(B)$ denote the number of irreducible c…

Broué's perfect-isometry conjecture. There should be a perfect isometry between $K_0(B)$ and $K_0(b)$.

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…

Let $G$ be a finite group and $B$ a block of $kG$ over a field $k$ of characteristic $p$, with defect group $D$. Benson's conjecture. The Loewy length of $B$ should satisfy … This…

Let $\pi$ be a set containing at least two primes, and let $\ell \in \pi$. If $G$ is a finite group whose order is divisible by $\ell$, a pi-prime-degree character is a character w…

Aschbacher–Kessar–Oliver's conjecture. If a fusion system $\mathcal{F}$ on $S$ is exotic, then $\mathcal{F}$ is block-exotic.

Let $G$ be a finite group, let $p$ be a prime, and let $B$ be a $p$-block of $G$ with defect group $D$. Write $\operatorname{ht}(B)$ for the set of heights of the irreducible chara…

Let $G$ be a finite group, let $p$ be a prime, and let $B$ be a $p$-block of $G$ with defect group $D$. Write $\operatorname{ht}(B)$ for the set of heights of the irreducible chara…