11 problems

Zassenhaus conjectures. If $H\in\mathcal{G}_i$, then there exists a subgroup $K_H\leqslant G$ such that $H$ and $K_H$ are conjugate in the unit group of ${\mathbb Q}G$.

Kimmerle's Prime Graph Question. Do the GK-graphs of $G$ and $\operatorname{V}(\mathbb{Z}G)$ coincide?

First Zassenhaus Conjecture. Every unit of augmentation one in $\mathbb{Z}G$ is conjugate to some group element by a unit of $\mathbb{Q}G$.

Hertweck's blockwise isomorphism conjecture. For every $e\in\operatorname{PCI}({\mathbb Q}G)$,

Blockwise Spectrum Problem. For every primitive central idempotent $e\in\operatorname{PCI}({\mathbb Q}G)$,

Jespers–Sun's indecomposability conjecture. The subgroup $\langle\mathcal{U}(\mathbb{Z}G)_{\mathrm{un}}\rangle$ is indecomposable.

Jespers–Sun's conjecture. A finite group $G$ has ND if and only if $\mathbb{Q}G$ has at most one matrix component.

Let $G$ be a finite group, and let ${\mathbb Z}G$ be its integral group ring. The normalized unit group consists of the units of augmentation one in ${\mathbb Z}G$; write it as…

Let $G=\operatorname{PSL}(2,p^f)$, and let $A_{p^f}$ be the subset of $\mathbb{N}$ consisting of those positive integers $n$ for which the HeLP method proves that every unit…

Let $G$ be a group, and let $u$ be a Bass cyclic unit of $G$ based on a non-central element of prime order. Assume that $u$ has infinite order modulo the centre of…

Let $G$ be a finite group, and let $u$ be a Bass unit based on an element of $G$ of prime order. Assume that $u$ has infinite order modulo the center of the units of the integral g…