University of Birmingham, Wednesday January 25th 2017

All lectures will take place Lecture Room B of the Watson Building.

**Owing to exceptional budget constraints we politely request that any postgraduate who will be requesting that we cover their travel expenses seek-out the cheapest means of getting here possible (typically a pre-booked train is fine).**

14:00-15:00 Carolina Vallejo Rodríguez (ICMAT) Detecting local properties in the character table

Let *G* be a finite group and let *p* be a prime number. In this talk, we discuss local properties of *G* that can be read off from its character table. More precisely, we characterize globally when the principal block of the normalizer of a Sylow *p*-subgroup has one simple module for *p* odd. We also talk about the *p*=2 case of this problem, which remains open. This is joint work with G. Navarro and P. H. Tiep.

15:00-16:00 Eugenio Gianelli (Cambridge) Characters of odd degree of symmetric groups

Let *G* be a finite group and let *P* be a Sylow *p*-subgroup of G.

Denote by Irr_{p‘}(G) the set consisting of all irreducible characters of G of degree coprime to *p*.

The *McKay Conjecture* asserts that |Irr_{p‘}(*G*)|=|Irr_{p‘}(N_{G}(*P*))|.

Sometimes, we do not only have the above equality, but it is also possible to determine explicit *natural* bijections (McKay bijections) between Irr_{p‘}(*G*) and Irr_{p‘}(N_{G}(*P*)).

In the first part of this talk I will describe the construction of McKay bijections for symmetric groups at the prime *p*=2.

In the second part of the talk I will present a recent joint work with Kleshchev, Navarro and Tiep, concerning the construction of natural bijections between IIrr_{p‘}(*G*) and Irr_{p‘}(*H*) for various classes of finite groups *G* and corresponding subgroups* H* of odd index. This includes the case* G=S _{n}* and

*H*any maximal subgroup of odd index in

*S*, as well as the construction of McKay bijections for solvable and general linear groups.

_{n}16:00-16:30 Break

16:30-17:30 Geoff Robinson (Aberdeen/Lancaster) On a subgroup introduced by J.Grodal

(report on ongoing joint work with J. Grodal). We will discuss the structure of the (normal) subgroup of a finite group* G* generated by the elements whose centraliser has order divisible by the prime *p*. This leads quickly to a study of an interesting generalization of Frobenius complements.

The abelianization of the associated quotient group plays a role in J. Grodal’s work on endotrivial modules.