23 problems

Benson's conjecture. All indecomposable summands of $V\otimes V^*$ have dimension divisible by $2$, except for the single summand isomorphic to the trivial representation $\Bbbk$;…

Let $G$ be a finite $p$-group. Define $\omega_a(G)$ as the minimum of $\dim \operatorname{H}^{*}(G;\mathbb{F}_p)/\mathfrak{p}$ over associated primes $\mathfrak{p}$ of…

Let $p$ be a prime and let $G$ be a finite $p$-group. The Loewy length $\mathsf{L}(G)$ is the nilpotency index of the Jacobson radical of the group algebra $\mathbb{F}_p[G]$, and…

Passman's conjecture. If

Boston's conjecture. The groups

Let $p$ be a prime, and consider the two non-isomorphic groups of order $p^5$ denoted by $H_3$ and $H_4$ in the cited source. Conjecture on Beauville groups of order $p^5$. For…

Schmid's cohomological conjecture. For every integer $n$, one has

Generalized Benson conjecture. If $(\mathcal{C},G')$ is an induced $p'$-symmetry of $V$, then every prime dividing the order of $G'$ is less than $p$.

Generalized Benson conjecture. Every prime dividing the order of $G'$ is less than $p$.

Let $P$ be a finite non-abelian $p$-group. Write $\Phi(P)$ for its Frattini subgroup, $Z(\Phi(P))$ for the center of that subgroup, and say that a module is cohomologically trivial…

Let $p$ be a prime number, let $G$ and $H$ be finite $p$-groups, and let $\mathbb{F}$ be a field of characteristic $p$. The group algebras $\mathbb{F}G$ and $\mathbb{F}H$ are the a…

Let $p$ be a prime, let $G$ be a finite $p$-group, and let $K(G,C_p)$ denote the module of relative $(G,C_p)$-Brauer relations. For a sub-quotient $P=K/N$ of $G\times C_p$, write…

Let $p$ be an odd prime and let $r\geq p-1$ be an integer. Let … be the finite $p$-group defined in the source. Depth-two conjecture. The mod-$p$ cohomology ring…

Let $G$ be a finite $p$-group, and write $M(G)=H_2(G,\mathbb{Z})$ for its Schur multiplier. Write $\operatorname{exp}(G)$ for the exponent of $G$. Exponent conjecture for finite…

Fix an integer $r$. A mod $p$ cohomology algebra means the graded algebra $H^*(G;\mathbb F_p)$ of a finite $p$-group $G$ with coefficients in $\mathbb F_p$. DGG's finiteness conjec…

Let $G$ be a powerful $4^*$-central $2$-group with the $\Omega$EP property, and let $d=\operatorname{rk}(\Omega_1(G))$. Write $\Lambda$ for the exterior algebra. The cohomology con…

Let $G$ be a finite $p$-group of fixed rank $r$. A subgroup $K$ is powerful $p^*$-central with the $\Omega$EP if it has those properties. The powerful $p^*$-central subgroup conjec…

Stable-cohomology top-degree conjecture. There exists a $G$-module $A$ with trivial action such that

Let $p$ be a prime, let $k$ be a field of characteristic $p$, and let $(G_i \mid i \geq 0)$ be a coclass family of finite $p$-groups. Cohomology stabilization conjecture. There exi…

Let $G$ be a non-cyclic finite $p$-group of order $p^n$, where $n\geq 3$. Write $\operatorname{Aut}(G)$ for the automorphism group of $G$, and let $|G|$ denote the order of a group…

Let $G$ be a finite $p$-group, let $M(G)$ denote its Schur multiplier, and let $\exp(H)$ denote the exponent of a group $H$. Exponent conjecture for Schur multipliers. The exponent…

Let $p$ be a prime, let $G$ be a finite $p$-group, and let $H\cong \mathbb Z_p$. Write $\tilde G=G\times H$, let $N(G,H)$ denote the relative Burnside kernel, and for a subquotient…

Bovdi's conjecture. Is it true that