(Pro-)fusion systems

Friday 15th September 2017, Lancaster Local organiser: Nadia Mazza

(Pro-)fusion systems is the second meeting of the Research Group Functor Categories for Groups (FCG). Introduced in the '70s, fusion systems are categories which model how non-conjugate subgroups in a Sylow p-subgroup of a given finite group can fuse, i.e. become conjugate, in the whole group. The study of fusion systems has led to significant advances and improvements of proofs in group theory, and also provided useful links with algebraic topology.

The focus of the meeting will be on the use of fusion systems in the local to the global theory of finite groups and the theory of profinite groups.

The venue for the talks is the Postgraduate Statistics Centre (PSC), lecture theatre A54. The timetable is as follows:

• 1.30-2.30: Markus Linckelmann (City), On automorphism groups of finite group algebras
• 2.40-3.40: Ellen Henke (Aberdeen), Fusion systems, localities and the classification of finite simple groups
• 3.40-4.10: tea break;
• 4.10-5.10: Geoffrey Robinson (Lancaster/Aberdeen), Realising Fusion Systems via Amalgams

Abstracts

Markus Linckelmann (City), On automorphism groups of finite group algebras

The automorphism group of a finite-dimensional algebra over an algebraically closed field is an algebraic group. By contrast, the automorphism group of a finite group algebra over a p-adic ring with finite residue field is a finite group. The structure of the automorphism group of a finite group algebra over a p-local domain with an algebraically closed residue field is largely unknown; it seems to be unknown in general whether this group is finite.

We identify a `large' subgroup of this automorphism group in terms of the fusion systems of blocks of finite group algebras as well as their Dade groups. The background motivation is the - to date open - question whether Morita equivalent block algebras have isomorphic defect groups and fusion systems. This is joint work with Robert Boltje and Radha Kessar.

Ellen Henke (Aberdeen), Fusion systems, localities and the classification of finite simple groups

Aschbacher announced a program to find a new and better proof of the classification of finite simple groups. As a foundation, he introduced a local theory of fusion systems, which is in many ways analogous to the local theory of groups. However, some constructions which are easy in groups are difficult or even impossible in fusion systems. There is some hope that such difficulties can be overcome by working with localities attached to fusion systems. Localities are group-like structures introduced by Chermak. I will give an introduction to the subject and present some recent results by Chermak and myself.

Geoffrey Robinson (Lancaster/Aberdeen), Realising Fusion Systems via Amalgams

We will discuss how to realise fusion systems on p-groups via repeated group amalgamations, and consider applications to group representations.