30 April 2025 — Padova

Exotic fusion systems

Valentina Grazian, Università degli Studi di Padova

Abstract

Fusion systems made their first appearance in a 2006 paper by Puig and have since then been investigated by many researchers around the world. A fusion system is a structure that encodes the properties of conjugation between \(p\)-subgroups of a group, for \(p\) any prime number. Given a finite group \(G\), it is always possible to define the saturated fusion system realized by \(G\) on one of its Sylow \(p\)-subgroups \(S\): this is the category where the objects are the subgroups of \(S\) and the morphisms are the restrictions of conjugation maps induced by the elements of \(G\). However, not all saturated fusion systems can be realized in this way: when this is the case, we say that the fusion system is exotic. An important research direction in volves the study of the behavior of exotic fusion systems (in particular at odd primes). In this talk we will present an overview of recent results concerning the classification of saturated fusion systems on certain families of finite \(p\)-groups, highlighting the developments on the understanding of exotic fusion systems at odd primes.