74 problems

Diaz–Park sharpness conjecture. For every $n\geq 1$,

Martino–Priddy conjecture. The groups $G$ and $H$ have isomorphic $p$-fusion systems if and only if

Solomon's conjecture. Every simple $2$-fusion system is either a Benson–Solomon fusion system or the $2$-fusion system of some finite simple group.

Let $\mathcal F$ be a fusion system on a finite $p$-group $S$, and let $\mathcal F^z$ be a set of representatives of the fully $\mathcal F$-centralised $\mathcal F$-conjugacy class…

A saturated fusion system is a category $\mathcal{F}$ together with a finite $p$-group $S$, whose objects are all subgroups of $S$ and whose morphisms are injective group homomorph…

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…

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

Parker–Semeraro's conjecture. Every corefree fusion system on $P$ can be realized by a group $H$ satisfying

Parshall–Semones conjecture. For all but finitely many exceptions, if $(G,S)\in\mathcal S$ and $\mathcal F$ is a saturated fusion system on $S$ with $O_p(\mathcal F)=1$, then

Block-fusion-system characterization conjecture. The block-realizable fusion systems are precisely the fusion systems that occur as the $p$-fusion systems of finite groups.

Let $\mathcal F=\mathcal F_{(1)}$ be the fusion system in the cited table, and let $\operatorname{Ind}(\mathcal F)$ denote its set of $\mathcal F$-indecomposable characters. The ch…

Oliver's Schur multiplier conjecture. $\ker(\mu_{\mathcal L})$ is isomorphic to a subgroup of the Schur multiplier of $\mathcal F$.

Existence conjecture. There are $p'$-reduced locally finite $p$-artinian groups that are not locally finite, strongly $p$-artinian groups, and there are fusion systems that are LF-…

Let $\mathcal{F}$ be a saturated fusion system on a finite $p$-group $S$ of order $p^e$, and let $\alpha$ be a compatible family. Write $s$ for the sectional rank of $S$, and let…

Serwene's conjecture. Let $\mathcal{F}$ be a saturated fusion system. For any finite group $G$ having an $\mathcal{F}$-block $b$, it follows that $\mathcal{F}$ is realizable.

Let $\mathcal{F}$ be a saturated fusion system. A classifying space associated to $\mathcal{F}$ is a space encoding its fusion-theoretic structure, denoted by $B\mathcal{F}$. Gener…

Equality conjecture. These two integers are equal:

Kessar–Linckelmann–Lynd–Semeraro weight conjectures. The following assertions should hold:

Permutation characteristic-class conjecture. Let $\mathcal S'(\mathcal F)$ be the subring of $H^*(|\mathcal L|_p^\wedge;\mathbb F_p)$ generated by the images in cohomology of all m…

Let $p$ be a prime, let $P$ be a finite $p$-group, and let $\mathcal F$ be a saturated fusion system on $P$. The fusion system $\mathcal F$ is exotic if it is not of the form…

Cantarero–Combariza conjecture. Every $\mathcal{F}$-invariant indecomposable character of $P$ is a summand of $\rho_P$.

Let $G$ be a finite group, $D$ a defect group, $B$ a block algebra, and ${\cal B}$ the corresponding block. Let ${\cal L}{\cal P}={\cal L}{\cal P}(B)$ be the pointed fusion system…

Let $G$ be a finite group, $D$ a defect group, $B$ a block algebra, and ${\cal B}$ the corresponding block. Write … for a finitely generated algebra $\Lambda$ over ${\cal O}$. The…

Let $S$ be a finite $p$-group and let ${\mathcal{F}}$ be a saturated fusion system on $S$. A representation of $S$ is ${\mathcal{F}}$-invariant if its restriction along every morph…

Let $F$ be an algebraically closed field of characteristic $p>0$, let $G$ be a finite group, and let $B$ be a block of $FG$ with defect group $D$ and fusion system…