Hello We welcome you all, give some information for the week and thank our sponsors.

Cheryl Praeger: Big questions of finite permutation groups – some answered, others open

Kamilla Rekvenyi: Orbital Diameter of Primitive Permutation Groups

Getting to know each other

Melissa Lee: Primitive permutation groups: more problems and open questions

Hongyi Huang: Base-two primitive permutation groups

Emily Hall: Almost elusive groups

Saul Freedman: The intersection graph of a finite simple group

Rebecca Waldecker: Permutation groups acting under constraints

Donna Testerman: Linear groups seen from different angles

Aluna Rizzoli

Colva Roney-Dougal

Eileen Pan: Finite groups of Lie type

Veronica Kelsey: A survey of base size and other numerical invariants

Luca di Gravina: Möbius function of finite classical groups

David Szabo: Finite subgroups of transformation groups C. Jordan proved in 1877 that every finite subgroup of $\mathrm{GL}_n(\mathbb{C})$ has a normal abelian subgroup of index bounded by a function of $n$ -- in short, these finite subgroups are `almost' abelian. It is natural to investigate whether an analogous statement holds for the finite subgroups of natural transformation groups like the birational automorphism group of an algebraic variety, or the diffeomorphism group of a compact manifold. Recent developments on the topic by A. Guld (2020) and Pyber--Csikós--E. Szabó (2022) gave a positive answer when `abelian' is replaced by `nilpotent of class at most $2$', and by `nilpotent' in the respective cases. We will briefly discuss why the nilpotency class has to be at least $2$ in both cases focusing on the common purely group theoretic ideas.

Inna Capdeboscq

Gareth Tracey: The Goldschmidt-Sims conjecture The Classification of Finite Simple Groups has led to substantial progress on deriving sharp order bounds in various natural families of finite groups. One of the most well-known instances of this is Sims' conjecture, which states that the order of a point stabiliser in a primitive permutation group has order bounded in terms of its smallest non-trivial orbit length (this was proved by Cameron, Praeger, Saxl and Seitz using the CFSG in 1983). In the meantime, Goldschmidt observed that a generalised version of Sims' conjecture, which we now call the \emph{Goldschmidt--Sims conjecture}, would lead to important applications in graph theory. In this talk, we will describe the conjecture, and discuss some recent progress. Joint work with L. Pyber.

Mandi Schaeffer-Fry: Conjecture-Cracking with the Classification: Some Applications, New and Old, of the CFSG

Noelia Rizo

Virgilius-Aurelian Minuță: Group graded algebras over G-graded G-algebras

Margherita Piccolo: Representation growth of semisimple profinite groups

Sesuai "Yash" Madanha: Average number of zeros of characters of finite groups

SP Madireddi: The Foulkes module

Teaser and poster, open end! 2 minutes teaser for a poster. Koushik Paul: Construction of Specht modules

Alice Niemeyer

Daniel Rademacher: Constructive recognition of classical groups

Rebecca Waldecker: Backtrack methods and canonical images

Farzaneh Gholaminezhad: The G-graph of the Gyrogroups

Friedrich Rober: Wreath Product Decompositions

Mun See Chang: Overview

Anna Sucker + Lucas Wollenhaupt: Computing the alternating and symmetric square representations of classical groups

Laura Voggesberger: On algebraic groups, their Lie algebras, and nilpotent pieces

John McHugh: On the image of the trivial source ring in the ring of virtual characters of a finite group

Eilidh McKemmie

Scott Harper. The generating graph: spread and domination The generating graph of a group has as vertices the nontrivial elements of the group and two vertices are adjacent if the elements generate the group. I will discuss the recent classification of the finite groups whose generating graph is connected (joint with Burness and Guralnick) and related work on surprisingly small total dominating sets for generating graphs of simple groups (joint with Burness). Time permitting, I will discuss related ideas for infinite simple groups.

Colva Roney-Dougal

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