Seminario Teoría de grupos 

7/2/2023

11:30 Aula Naranja ICMAT

Speaker: Leo Margolis (UAM)

Title: Units in group rings and representation theory of groups of prime order

Abstract: The unit group $U(mathbb{Z}G)$ of the integral group ring $mathbb{Z}G$ of a finite group $G$ is a long studied object, but nevertheless many fundamental questions remain open. The strongest possible expectation on finite subgroups of $U(mathbb{Z}G)$, expressed by Zassenhaus, had been that these always lie inside the trivial units $pm G$ up to conjugation in the bigger group algebra $mathbb{Q}G$. While this has been refuted in this most general form, it is still open for $p$-subgroups, and also whether the order of elements in $U(mathbb{Z}G)$ coincide with those in $G$ remains unknown (after a suitable normalisation process).

I will explain a method which has been used to achieve several results on these questions, but the bottle neck of which, for the moment, is the understanding of representations of the most simplest groups one can imagine - the cyclic groups of prime order $p$. Overcoming some of these difficulties, recent progress allows us to show that Zassenhaus' question has a positive answer at least for units of order $p$, when the Sylow $p$-subgroup of $G$ is also assumed to be of prime order.

This is joint work with Florian Eisele.