Talks and Abstracts

The opening talk at the conference will be given by Prof. Radha Kessar, whilst Prof. Andrea Lucchini will give the closing talk.


Nilpotence and character degrees. - Prof. Radha Kessar
One of the major themes of group theory is the interplay between the representations of finite groups and their p-local structure. My talk, which will be mostly expository, will focus on the special situation of p-nilpotent groups and nilpotent p-blocks. I will also report on some recent joint work with M. Linckelmann and G. Navarro.


Some questions arising from the study of the Product Replacement Algorithm - Prof. Andrea Lucchini
The product replacement algorithm (PRA) is a practical algorithm to construct random elements of a finite group. The analysis of the efficient of this algorithm motivate a series of questions concerning the generation of finite and profinite groups. We will discuss some recent developments of these questions (maximal size of an independent generating set, connectivity of the generating graph and Swap conjecture, bias of group generators).

Student Talks and Posters

Tuesday 24th June

• 11.30-11.55: Julian Brough [Cambridge University]
  Extending the Notion of an Abelian Sylow Subgroup
  Let G be a nite group and p a prime divisor of the order of G. If G has an abelian Sylow p-subgroup, any p-element g will contain a Sylow p subgroup in its centraliser CG(g). If for all p-elements g there exists a Sylow p-subgroup of G contained in CG(g), does this imply that the Sylow p-subgroups are abelian? The Tits' group (2^F4(2)') provides a counter example to this question. All 3-elements contain a Sylow 3-subgroup in their centraliser, but the Sylow 3-subgroups are non abelian. We will define a group to be cl_p if all p-elements have a Sylow p-subgroup in the centraliser and the Sylow p-subgroup is non abelian. The aim of the talk is to highlight why there are very few cl_p groups.




• 12.00-12.25: Ainhoa Iniguez [Mansfield, University of Oxford]
  Words in finite groups
  Let G be a finite nilpotent group. Alon Amit conjectured that for a word w in k variables the number of k-tuples in G^{(k)} satisfying w, N(G,w), is at least |G|^{k-1}. It is easy to see that this bound is exactly reached for finite abelian groups. N. Nikolov and D. Segal define e(G) = p^{-(|G|-t)} where p is the largest prime divisor of |G| and t is the number of prime divisors of |G| and they show that for any finite nilpotent group G, inf_{w,a} N(G,w = a) > e(G)|G|^k where w ranges over F_{\infty} and a ranges over the w-values G_w^+. Motivated by this, Carolyn Ashurst conjectured that for any finite nilpotent group, inf_{w,a} N(G,w = a) = |G|^{k-1}. In this paper we show that if G is a finite group of nilpotency class two and w is a word in k variables, for any non-empty fibre w*{-1}(a), N(G,w = a) is at least |G|^{k-1}. This proves the conjectures of A. Amit and C. Ashurst for finite groups of nilpotency class at most two. A. Amit’s conjecture for finite groups of nilpotency class at most two was also proved independently by M. Levy.




• 14.00-14.25: Glen Collins [University of Birmingham]
  On Hall normally embedded subgroups
  Characterising groups which possess subgroups of a prescribed order, perhaps with further properties such as being normal, has long been a source of strong and useful results within group theory. A classic example of such a result is P. Hall's generalisation of Sylow's theorem to soluble groups. In this talk I will outline another characterisation result. Let $G$ be a group such that for any subgroup $B\leq G$, there exists a Hall normally embedded subgroup $H\leg G$ such that $|B|=|H|$. It was recently asked by S. Li and J. Liu whether such groups are soluble. I have managed to answer this question in the affirmative and have given a precise description of the structure of these groups. I will also talk about how this characterisation result relates to characterisations of other classes of groups.




• 14.30-14.55: Louise Sutton [Queen Mary, University of London]
  The graded dimension of Specht modules
  Representation theory of the symmetric groups has been an area of interest and extensively studied for many years. The problem of fully understanding the symmetric groups over a field of positive characteristic remains at the forefront of representation theory. In 1954, the hook length formula determining the dimension of Specht modules was proven. Until the symmetric group algebra was shown to be isomorphic to a certain KLR algebra in 2009, it was unknown how to grade the symmetric group algebras. We can then define the graded dimensions of Specht modules by Laurent polynomials. Ambitiously, one can hope of obtaining a graded hook length formula. This talk gives an overview of the KLR algebra, together with my recent work on the graded dimensions of the Specht modules for the symmetric group.




• 15.00-15.25: Ihechukwu Chinyere [Heriot-Watt University]
  Solution of singular equation over free product
  Given two nontrivial groups A and B and a word r=atbt^{-1}ctdt^{-1} in A*B* \langle t \rangle (where a, c are in A and b, d in B), we show that the equation r^n=1 has a solution over A*B where n is in \mathbb{Z}-\lbrace 1 \rbrace i.e the natural map from A*B to G=(A*B* )/(\langle \langle r ^ n \rangle \rangle) is injective.




• 16.00-16.25: Henry Bradford [St John's College, Oxford]
  Uniform Diameter Bounds for Families of Finite Groups
  I shall outline a general method for finding upper bounds on the diameters of finite groups, based on the Solovay-Kitaev procedure from quantum computation. This method may be fruitfully applied to groups arising as quotients of many familiar pro-p groups. Time permitting, I will indicate a connection with weak spectral gap, and give some applications.




• 16.30-16.55: Ben Quigley [Dublin City University]
  
Regarding the presentation of the commutator subgroups of the Artin groups B3 and F4
We examine the Artin group B3 and its commutator subgroup. We use the generating set of non-crossing partitions to construct a complex which is a K(p,1) for the commutator subgroup. We use this K(p,1) to recover the fact that the commutator subgroup is finitely generated and to show that it is not finitely presented. This is an example of a general construction which should also extend to F4.

Wednesday 25th June

• 10.00-10.25: Raul Moragues Moncho [University of Warwick]
  
Algorithms for subgroups of free groups
The Todd-Coxeter algorithm for coset enumeration gives a lot of information about finite index subgroups of arbitrary groups, but does not terminate for infinite index subgroups. In the case of finitely generated subgroups of infinite index of free groups, despite the algorithm not terminating, it is still possible to recover a lot of information from the incomplete coset table. Using this, I wrote algorithms that allow to intersect two such subgroups, decide whether a word is in a given subgroup, calculate the normaliser of a subgroup and decide whether two subgroups are conjugate, using the Magma algebra system.



• 10.30-10.55: Emilio Pierro [Birkbeck, University of London]
  
New Examples of Mixed Beauville Groups
Beauville Surfaces are a class of rigid complex surfaces of general type. Associated to a Beauville surface is a group, known as a Beauville group, which comes in both mixed and unmixed flavours. Through the work of various mathematicians it has recently been shown that, with the exceptions of $A_5$ and $SL_2(5)$ all simple and quasi-simple groups are unmixed Beauville groups. Mixed Beauville groups, however, have proven much harder to find. We generalise a construction of mixed Beauville groups first given by Bauer, Catanese and Grunewald, then go on to give several examples of infinite families of characteristically simple groups that satisfy the hypotheses of our theorem and thus provide a wealth of new examples of mixed Beauville groups.



• 11.30-11.55: Elisa Covato [University of Bristol]
  
Fixed point spaces for algebraic group actions
The fixed point ratios are widely studied in the context of finite permutation groups. In this talk, we will discuss their analogous, the fixed point spaces, for algebraic groups. In particular, we will focus our attention on the study of the fixed point spaces of involutions in symplectic groups.



• 12.00-12.25: Matthew Spencer [University of Warwick]
  
Brauer Relations over fields of positive characteristic
To any set X equipped with an action of some finite group G we may associate a representation of G over a field K. In this process different G-sets may give rise to the same representations, and we call such relations Brauer relations. In my talk I will introduce a convenient language to describe such relations, as well as give examples of such relations for Char(K)=0,p>0 and state some results concerning which groups admit relations.



• 14.00-14.25: Amit Hazi [Cambridge University]
  
Tilting modules for algebraic groups
Let G be a semisimple algebraic group. A G-module T is called a tilting module if it has a filtration whose quotients are Weyl modules and another filtration whose quotients are dual Weyl modules. Tilting modules are interesting in this context because they appear as tensor products of irreducible modules. In this talk I'll discuss more about how tilting modules arise, some recent developments in calculating their structure in small cases, and some conjectures which could speed this process along.



• 14.30-14.55: Martin Hauge [University of Bristol]
  
Varieties for group modules
Cohomology is an indispensable tool in the modular representation theory of finite groups. The cohomology ring of a finite group is close enough to being a commutative ring that we can regard it as the coordinate ring of an algebraic variety, and we can assign subvarieties to modules in a nice way. I'll explain this construction, and how the varieties can in fact be computed easily, without knowing the cohomology of the group. We'll also see how varieties classify certain nice subcategories of the module category of a p-group.

Thursday 26th June

• 10.00-10.25: Charles Cox [University of Southampton]
  
Dehn's Problems and Houghton's groups
Dehn's Problems resulted in some of the first undecidability results outside of set theory and logic. We shall introduce these problems and solve The Word Problem (and possibly The Conjugacy Problem) for Houghton's groups. These include a family of finitely presented groups which each contain the infinite symmetric group.



• 10.30-10.55: Oliver King [City University London]
  
Modular representation theory of diagram algebras
The Brauer and partition algebras, introduced by Brauer and Martin respectively, are examples of diagram algebras. The representation theory of both of these over a field of characteristic zero is well understood. In this talk, I will recall the block structure of the Brauer and partition algebras in characteristic zero in terms of the action of a reflection group on the set of simple modules. I will then give a description of the blocks in positive characteristic by using the corresponding affine reflection group (for the partition algebra, this is joint work with C. Bowman and M. De Visscher).



• 11.30-11.55: Simon Peacock [University of Bristol]
  An introduction to separable equivalence
  There are several notions of algebraic equivalence: isomorphism, Morita equivalence, stable equivalence, derived equivalence and so on. I will briefly talk about some of these and introduce the idea of separable equivalence of algebras. This relation can be thought of as a generalization(ish) of the previously mentioned equivalences. If I have time I will also tell you something more than just the definition... though I make no promises.




• 12.00-12.25: Imke Toborg [University of Koblenz-Landau]
  
Local Methods for a Theorem of $Z^*_3$-type
Is it possible to prove a generalisation of Glauberman's Z*-theorem to odd primes without using the classification of finite simple groups? In this talk I will present some ideas of local arguments for a new proof of Peter Rowley's result that 3-locally central elements of a finite group are contained in Z*3(G) for a finite group G which is a weaker version of the desired statement.



• 14.00-14.25: Alec Gullon [Lancaster University]
  
Introduction to the group of endotrivial modules
We introduce the group of endotrivial modules, an important object in the field of modular representation theory, and discuss some known results concerning it.



• 14.30-14.55: Gareth Tracey [University of Warwick]
  
How many elements does it take to generate a finite permutation group?
For a group G, let d(G) denote the minimal number of elements required to generate G. When G is a finite permutation group of degree d, one can bound d(G) in terms of d, but how sharp can this bound be? Does the bound improve much when we restrict our attention to special classes of permutation groups, like primitive or transitive groups? These questions were first looked at in the late 1980s, after the classification of finite simple groups was announced, and the recent trend has been to come up with results of the form: ``there exists a constant C such that, for each (primitive/transitive/soluble/nilpotent/etc) permutation group G of degree d, we have d(G)\le Cf(d)", where f is some function of d. However, the constants involved were never estimated. In this talk, we will go through some of these results, and we will also look at ways to estimate the constant C.



• 15.00-15.25: Katherine Horan [University of Kent]
  
When are Rings of Invariants of Finite Groups Polynomial Rings?
Suppose that we have a vector space V and a finite group G contained in GL(V), acting on a polynomial ring K[V], K a field of characteristic p. One of the questions we can ask is ``when is the ring of invariants K[V]^G = {f \in K[V] | g(f)=f, for all g in G} also a polynomial ring?’’ How far away is it from being polynomial otherwise? In this talk I aim to discuss when the ring of invariants is polynomial in the non-modular case, and what is known in the modular case. I will also discuss general structural properties of rings of invariants in cases where they are not known to be polynomial.



• 16.00-16.25: Ged Corob Cook [Royal Holloway, University of London]
  
Profinite tensor products
Given a ring $R$, the tensor product of two $R$-modules can be defined by a middle linear map. If $R$ and the modules also have a topology, one can define a topological tensor product by considering only continuous middle linear maps. In the special case where these topologies are profinite we give necessary and sufficient conditions for the profinite tensor product to be isomorphic to the usual (abstract) one. Finally we sketch to comparing profinite $Tor$ groups to their abstract equivalents.



• 16.30-16.55: John Rizkallah [Cambridge University]
  
On the cohomology of SL_2 with coefficients in an irreducible module
Let G be the simple algebraic group SL_2 defined over an algebraically closed field of positive characteristic, and let M be an irreducible G-module. I will discuss recent work which seeks to determine an upper bound for the dimension of the cohomology space H^q(G,M).

Friday 27th June

• 10.00-10.25: William Wong [City University London]
  
Representation theory as group automorphisms
Group automorphisms is a great tool to study the structure of groups. However in some cases looking at them is not enough. In this talk, we will introduce some further automorphisms in algebras that generalize group automorphisms, and see very briefly what we can do with it.



• 10.30-10.55: Valentina Grazian [University of Birmingham]
  
Saturated Fusion Systems over p-groups of sectional rank at most 3
A saturated fusion system over a finite p-group S is a category whose objects are the subgroups of S and whose morphisms are the monomorphisms between subgroups which satisfy certain axioms, first formulated by Puig and motivated by conjugacy relations among p-subgroups of a given finite group. In the first part of my talk, I will give an introduction to Fusion Systems, with some examples. Next, I will talk about the classification of Saturated Fusion Systems over p-groups of sectional rank at most 3 for p odd, giving some new results.