60 problems

Let $\Gamma$ be a torsion-free group, and let $\mathbf{C}\Gamma$ be its group algebra over the complex numbers. A zero divisor is a nonzero element that annihilates another nonzero…

Let $G$ be a torsion group and let $\mathbf{F}G$ be its group algebra over a field $\mathbf{F}$. Suppose that the unit group $\mathcal{U}(\mathbf{F}G)$ satisfies a group identity.…

Donald–Flanigan conjecture. The group algebra $kG$ admits a $k((t))$-separable deformation.

Passman's conjecture. If

Let $d$ be a negative integer. The group ${\rm SL}_2({\mathbb Z}[\sqrt{d}])$ is virtually free-by-free if it has a free subgroup with free quotient of finite index. Virtual free-by…

Analytical zero-divisor conjecture. If $\Gamma$ is torsion-free, then $\lambda_{\Gamma}(f)$ is injective for every non-zero $f\in\mathbf{C}\Gamma$. This is a classical conjecture c…

Cameron's prime-element conjecture. If $G$ has no finite orbits, then $\varepsilon$ is a prime element in $A(G)$.

Let $G$ be a finite group, let $k$ be a field whose characteristic divides the order of $G$, and let $B$ be a block of the group algebra $kG$ with defect group $P$. Linckelmann's c…

Finite-Krull-dimension conjecture. The following assertions hold:

Noetherian group-algebra conjecture. If $kG$ is noetherian, then $G$ is polycyclic-by-finite.

Semiperfectness criterion. The ring $\mathbb{Z}_{(p)}G$ is semiperfect if and only if

Commutativity conjecture. For fixed $n$, the elements $\mathbf{X}_{a,b}$ commute for all $a,b\in\mathbb{N}$; equivalently,

Let $G$ be a group, let $H\subseteq G$ be a subgroup of finite index satisfying an additional group-theoretic condition, and let $M$ be a module over the group algebra of $G$. Moor…

Let $G$ be a group and let $k$ be a commutative ring of finite global dimension. Write ${\tt Cof}(kG)$ for the class of cofibrant $kG$-modules and ${\tt GProj}(kG)$ for the class o…

Dihedral group-algebra length conjecture.

Let $\mathrm{G}$ be a finite group, let $p$ be a modular prime for $\mathrm{G}$, and let $\mathcal{L}_p[\mathrm{G}]$ be the associated modular Plesken Lie algebra. Let…

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…

The p-hyperfocal subgroup conjecture. The group algebra $kG$ is $\tau$-tilting finite if and only if either $R$ is cyclic, or $p=2$ and $R$ is isomorphic to a dihedral, semidihedra…

Universal Hughes-free embedding conjecture. Group algebras of all locally indicable groups have Hughes-free embeddings that are universal in a suitable sense.

Kaplansky's zerodivisor conjecture. The group algebra $kG$ is a domain.

Let $G$ be a finite group, and let $\mathbb{C}[G]$ be its complex group algebra. A bialgebra is of Knutson type if every simple module over it has trivial Knutson index, meaning th…

Vershik's conjecture. The growth of this associated graded algebra is subexponential.

Fix $\varepsilon>0$. Let $G$ be a finite group of order $n$, and say that $G$ is $n^\varepsilon$-quasirandom when every nontrivial irreducible complex representation of $G$ has dim…

Let $G$ be a finite group, let $M(G)$ denote its matching number, and define the minimum slice rank … where the minimum is over algebraically closed fields $k$ and $T_{k[G]}$ is th…

Strong Arens irregularity conjecture. $L^1(G,\mathcal A)$ is S.A.I. if and only if $\mathcal A$ is S.A.I..