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 ${\mathbb Q}G$.

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

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.…

Probabilistic subgroup-generation conjecture. Even though one does not generally know that $H=K_S$, the kernel and cokernel of $\phi$ should be the unit group and ideal class group…

Let $G$ be a finite group and let $H\leq\operatorname{U}(\mathbb{Z}G)$ be a subgroup of the normalized unit group of the integral group ring. The strongest Zassenhaus conjecture. T…

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

Nontrivial-unit conjecture. If $\deg g>\deg h$ and $h\ne 0$, then

Let $K$ be a non-big field, meaning an algebraic extension of $\mathbb{Q}$ that is not big, and let $R_K$ denote the ring constructed from the unit group of $K$. Triviality conject…

Let $A$ be a ring of type $2$, let $p$ be a prime satisfying $p\equiv3\pmod4$, and let $(A^*)_p$ denote the Sylow $p$-subgroup of the unit group of $A$. Consider a finite abelian…

Shapes of unit lattices conjectures. The closure $\overline{\Omega}$ in the modular surface is non-compact, the closure $\overline{\Omega}$ in the modular surface has non-empty int…

Let $a$ be an integer and let $\alpha$ be a root of … The field $\mathbb{Q}(\alpha)$ is a Shanks simplest cubic field, and $\mathbb{Z}[\alpha]$ is its associated order. Writing…