Finite groups in integral group rings
Ángel del Río
TL;DR
The note surveys the structure of finite subgroups and torsion units in the integral group ring $\mathbb{Z}G$, grounding the investigation in the Berman–Higman framework and the augmentation-based decomposition $\mathcal U(\mathbb{Z}G)\cong \{\pm 1\}\,V(\mathbb{Z}G)$. It introduces partial augmentations and the double-action formalism to reformulate the Zassenhaus problems as module- or representation-theoretic questions, and presents the HeLP method as a practical, character-theoretic tool to constrain possible torsion units and verify conjectures in specific families (notably $\mathcal A_5$). The text also connects these questions to broader isomorphism and automorphism problems for group rings and surveys positive/negative results in that landscape. A major highlight is Hertweck’s result that the Spectrum Problem holds for solvable groups, linking the spectra of $G$ and its unit group $V(\mathbb{Z}G)$ via deep modular and homological techniques. Overall, the work synthesizes representation theory, augmentation techniques, and computational methods to advance understanding of torsion units and their relation to the underlying group $G$.
Abstract
Notes used for a course held in 2016 in the School of Advances in Group Theory and Applications, for some lectures given in 2018 for the students of the Master in Mathematics of the Vrije Universiteit Brussels, a course for master and Ph.D. students at the Universidade de São Paulo and at the conference Group algebras, representations and computations. We revise some problems on the study of finite subgroups of the group of units of integral group rings of finite groups and some techniques to attack them.