6 problems

Multiplicative Burnside conjecture.

Let $G$ be a finite group and let $n\geqslant0$. The Ravenel ideal $\mathrm{J}_n(G)$ consists of virtual finite $G$-sets whose fixed-point sets have virtual cardinality zero for ev…

Let $G$ be a finite group, and let $\beta(G)$ denote its largest quotient that is a $B$-group. Thévenaz's conjecture. The group $\beta(G)$ is solvable if and only if $G$ is solvabl…

Deviation conjectures. With $\Delta=\Delta(G,H)$, one has $\Delta=1$ if and only if $H=G$, the index $[G:H]$ divides $\Delta$, and $\Delta$ divides $|G|$. These conjectures propose…

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…

Let $G$ be a finite group. Let $\Omega(G)$ be the Burnside ring, let $\Omega^*(G)$ denote its unit group, and let $\Omega_2(G)$ be the subring of monomial representations induced f…