15th Meeting: Finite Groups

University of Manchester, Wednesday May 23rd 2018

Venue: Room G108 of the Alan Turing Building, University of Manchester.

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 – note that most people in this category are elgibile for a rail-card).

13:00-13:50 Raul Moragues Moncho (University of Birmingham) Fusion systems over p-groups with an extraspecial subgroup of index p

Saturated fusion systems are constructions on p-groups which generalise the action of a finite group on its Sylow p-subgroups via conjugation. They have uses in representation theory and topology, but we focus on finite simple groups. Starting from a p-subgroup, we introduce homomorphisms between its subgroups which behave like conjugation by an element of an overgroup. There arise exotic configurations which do not come from finite groups, one of which was found in the search for the finite simple groups.

In order to find and study new exotic configurations, fusion systems have been classified over certain families of p-groups. We focus on p-groups S with an extraspecial subgroup of index p when p is odd, which appear as Sylow p-subgroups of PSL4(p), PSU4(p), PSp4(p) and G2(p) if p is large enough, as well as some other finite simple groups when p is small. We present a classification of fusion systems over S with no normal p-subgroups, which includes some new exotic examples related to a family constructed by Parker and Stroth.

14:00-14:50 Scott Harper (University of Bristol) 3/2-Generation of Finite Groups

It is well known that every finite simple group can be generated by two elements. Moreover, two arbitrary elements are very likely to generate the entire group. For example, every non-identity element of a finite simple group belongs to a generating pair. Groups with the latter property are said to be 3/2-generated. It is natural to ask which other finite groups are 3/2-generated. In 2008, Breuer, Guralnick and Kantor conjectured that a finite group is 3/2-generated if and only if every proper quotient of the group is cyclic.

In this talk, I will introduce this area of research before reporting on recent progress towards establishing this conjecture, where probabilistic techniques play a key role. I will also highlight connections with related topics, such as recent work on generating graphs.

15:00-15:45 Break

15:45-16:35 Sarah Rees (University of Newcastle) Negative curvature in some Artin groups

Artin groups are those groups with presentations of the form
< x1, x2, …, xn | xixjxi…= xjxixj… where each word has length mij, 0<i,j<n+1>
where mij are at least 2 and possibly infinite. Unless an Artin group is free, it
can’t be hyperbolic because it contains subgroups isomorphic to the free abelian group of rank 2. But despite  that, it’s been known for more than 30 years that many Artin groups (certainly all those of ‘large’ type, for which no pairs of generators commute, or rather,  mij>2, for all i,j) exhibit a weak form of relative hyperbolicity. Recent work of Huang and Osajda now proves that all those Artin groups, and a few more, are ‘systolic’, that is, they act geometrically on ‘systolic complexes’. Hence, in particular, they are all biautomatic, and display various algorithmic properties which are always found in hyperbolic groups.

This is consistent with my own relatively recent work with Holt, which proved that all these Artin groups (and more) are shortlex automatic, with effective rewrite systems to a (shortlex minimal) normal form. I’ll talk about ongoing work with Holt, to try and explain how these quite different approaches give many of the same answers.

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14th meeting: Finite Groups

University of Bristol, Wednesday February 7th 2018

Venue: Lecture theatre SM1, School of Mathematics, University of Bristol

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 – note that most people in this category are elgibile for a rail-card).

14:00-15:00 Carlisle King (Imperial College, London) Probabilistic methods and generation of finite simple groups

Let G be a finite group. It is well-known that G is generated by a pair of elements – we say G is 2-generated. Later, it was shown that 2 elements chosen at random from G generate G with probability tending to 1 as |G| tends to infinity. A natural refinement is to ask, given a pair of integers (a,b), whether G is 2-generated by an element of order a and an element of order b. If such a pair exists, we say that G is (a,b)-generated. We will explore some past results regarding (2,3)-generation proved using probabilistic methods, as well as a recent result on (2,p)-generation for some prime p.

An equivalent statement of the 2-generation theorem is that every finite simple is an image of F2, the free group on 2 generators. More generally, given a finitely presented group Γ, one can ask which finite simple groups are images of Γ. We will study a new result in this area concerning the free produc Γ=A*B of nontrivial finite groups A and B, also proved using probability methods.

15:00-15:30 Break

15:30-16:30 Stacey Law (Cambridge) p’-branching for symmetric groups

Let p be any prime and n be any natural number. Let χ be an ordinary irreducible character of the symmetric group Sn whose degree is coprime to p.

We bound the number of p’-irreducible constituents of the restriction of χ to Sn−1. This generalizes work of Ayyer, Prasad and Spallone (2016) for the prime p = 2. This is joint work with E. Giannelli and S. Martin.

16:30-17:30 Nick Gill (South Wales) Cherlin’s conjecture, Lachlan’s theory of homogeneous relational structures and the notion of “sporadicness”

Some years ago, Lachlan advanced a theory on homogeneity in relational structures which imposed a natural “hierarchy of complexity” on the universe of homogeneous relational structure. This theory was reworked by Cherlin in the 1990’s with a view to understanding finite permutation groups from a model theoretic point of view.

One upshot of all this is that we know the existence of an infinite family of theorems describing the so-called “relational complexity”  of all finite permutation groups. The problem is that, although we know these theorems exist, and even have a “form” for them, nonetheless we do not yet have the precise statement of any of them. However Cherlin has conjectured what (part of) the first of these theorems should say, and we will discuss this conjecture at some length.

There has also been substantial progress on this conjecture due to Cherlin himself, to Wiscons, and to myself and various co-authors. In particular the recent results that I will describe are due to myself, Pablo Spiga, Francis Hunt and Francesca Dalla Volta.

The talk has elements of model theory, combinatorics and finite permutation groups, and should be accessible to all.

13th meeting: Groups & Geometry

SUBJECT TO THE LMS AWARDING FUNDING TO THE “TRIANGLE” NEXT YEAR…

Birkbeck, University of London, Wednesday December 13th 2017

Room B13 of 43 Gordon Square. Note that this is not the usual building but is nearby (if you’ve walked to Birkbeck from somewhere like Euston or King’s Cross before then you’ve practically walked past it. You’re warned that an accurate map of the building itself can be found here.) If you would like some directions then email b.fairbairn[usual funny AT symbol]bbk.ac.uk . Afterwards we will very likely go The Bree Louise followed by a curry (assuming they’re still standing – like Brexit and Winter, HS2 is coming…)

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).

13:30-14:30 Emilio Pierro (LSE) Big groups and their small quotients

In this talk we will discuss new techniques in determining which finite simple groups can and cannot occur as quotients of certain infinite groups. Our motivation is to prove a conjecture of Mechia-Zimmerman stating that Ln(2) is the smallest non-trivial quotient of Aut(Fn), the automorphism groups of the free group of rank n. Our techniques can also then be applied to solve the analogous conjecture of Zimmerman that S2g(2) is the smallest non-trivial quotient of MCG( Σg,b), the mapping class group of a connected, orientable surface of genus g>2 with b non-negative boundary components and no punctures. This is joint work with Barbara Baumeister and Dawid Kielak.

14:30-15:30 Henry Bradford (Göttingen) Short Laws for Finite Groups and Residual Finiteness Growth

A law for a group G is a non-trivial equation satisfied by all tuples of elements in G. We study the length of the shortest law holding in a finite group. We produce new short laws holding (a) in finite simple groups of Lie type and (b) simultaneously in all finite groups of small order. As an application of the latter we obtain a new lower bound on the residual finiteness growth of free groups. This talk is based on joint work with Andreas Thom.

15:30-16:00 Break

16:00-17:00 Caroline Series (Warwick) Geometry in non-discrete groups of hyperbolic isometries: Primitive stability and the Bowditch condition are equivalent

It appears that there can be geometrical conditions on a group of hyperbolic isometries which may sometimes be of interest even when the group is not discrete.  We explain two different such conditions which pertain to the primitive elements in an SL(2,C) representation of the free group F2.  One is Minsky’s condition of primitive stability, and the other is the so-called BQ-condition introduced by Bowditch and generalised by Tan, Wong and Zhang. We prove these two conditions are equivalent, aided by an auxiliary condition which constrains the  location of the axes of those primitive elements which are palindromic words.

12th meeting: Permutation Groups

University of Bristol, May 17th 2017

School of Mathematics, University Walk, lectures will take place in room SM2.

After the last talk we will go for drinks and dinner.

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 Joanna Fawcett (University of Cambridge) Finite k-connected-homogeneous graphs

A graph is homogeneous if any isomorphism between finite induced subgraphs extends to an automorphism of the entire graph. The finite homogeneous graphs have been completely classified, and only a few families of examples arise. In this talk, we will discuss several methods of relaxing the hypothesis of homogeneity. In particular, we will discuss some recent progress on classifying the finite k-connected-homogeneous graphs where k>3. This is joint work with A. Devillers, C.H. Li, C.E. Praeger and J.-X. Zhou.

15:00 – 15:30 Break

15:30 – 16:30 Melissa Lee (Imperial College) Bases of quasisimple linear groups and Pyber’s conjecture

A base of a group G acting faithfully on a set  Ω is a subset B⊆Ω such that the pointwise stabiliser of B in G is trivial. The minimal base size of G is denoted by b(G).

A well-known conjecture made by Pyber in 1993 states that there is an absolute constant c such that if G acts primitively on Ω, then b(G) < clog|G|/log n, where |Ω|=n.

Following the contributions of several authors, the conjecture was finally established in 2016 by Duyan, Halasi and Maróti.

A result that played a major role in the proof of Pyber’s conjecture for primitive linear groups was given by Liebeck and Shalev, who proved that there is a constant C such that if G is a quasisimple group acting irreducibly on a finite vector space V, then either b(G)C, or G is an alternating or classical group acting on its natural module. In this talk, I will cover the history of Pyber’s conjecture, especially in the context of primitive linear groups, and present some results on the determination of the constant C for bases of quasisimple groups. I will also discuss an application by Liebeck of the latter result, which improves the known upper bounds for b(G) when G is an irreducible primitive linear group.

16:30 – 17:30 Simon Smith (University of Lincoln) The box product of two permutation groups

There are a number of ways in which one may take the product of two groups. Products which possess some kind of “universal” property (like the free and wreath products), or those which preserve some of the important properties of the input groups, are rare and precious.

Arguably, the most important product in permutation group theory is the wreath product, acting in its product action. The reason for this is that, unlike other products, it preserves a fundamental property called primitivity. Primitive permutation groups are indecomposable in some sense, and for finite groups they are the basic building blocks from which all permutation groups are comprised.

I am going to talk about a new product, called the box product. It is fundamentally different to the wreath product in product action. Nevertheless, it preserves primitivity under astonishingly similar conditions. Moreover, the box product has a “universal” property, and under natural conditions on groups M and N, the product of M and N is simple. The product can be used to easily solve a well-known open problem from topological group theory, and has an important role to play in the classification of infinite permutation groups.

11th meeting: Finite Groups

University of Manchester, Wednesday March 1st 2017

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).

All talks will be held in Room G205 of the Alan Turing Building

1.30 – 2.20 Sarah Hart (Birkbeck)  Product-free sets and Filled Groups

2.30 – 3.20 Ali Aubad (Manchester) Commuting Involution Graphs for Double Covers of the Symmetric Groups

3.30 – 4.00   Refreshments

4.00-4.50 David Ward (Manchester) Cuspidal Character and Finite Sporadic Simple Groups

The last talk will be followed by going for further refreshment followed by a curry.

10th meeting: Groups & Representations

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 Irrp(G) the set consisting of all irreducible characters of G of degree coprime to p.

The McKay Conjecture asserts that |Irrp(G)|=|Irrp(NG(P))|.

Sometimes, we do not only have the above equality, but it is also possible to determine explicit natural bijections (McKay bijections) between Irrp(G) and Irrp(NG(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 IIrrp(G) and Irrp(H) for various classes of finite groups G and corresponding subgroups H of odd index. This includes the case G=Sn and H any maximal subgroup of odd index in Sn, as well as the construction of McKay bijections for solvable and general linear groups.

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.

9th meeting: Groups & Combinatorics

Birkbeck, University of London, Wednesday December 14th 2016

All talks will be held in room B04 of the main Malet Street building. Directions available on request from b.fairbairn [at ] bbk.ac.uk if needed.

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).

13:30-14:30 Atiqa Sheikh (Imperial) Orbital diameters of the alternating and projective special linear groups

In this talk we explore a particular family of edge-transitive graphs called orbital graphs. The orbital diameter of a primitive group G acting on a finite set Ω is defined to be the maximum of the diameters of the orbital graphs of G. We will give an overview of the results obtained on the orbital diameters of groups whose socle is an alternating group or a projective special linear group of dimension 2. The results concern boundedness of the orbital diameters and classifying orbital graphs of small diameters.

14:30-15:30 Jozef Širáň (OU) Regular maps arising from twisted linear groups

A map a cellular embedding of a graph on a surface. A flag of a map is a triple consisting of a vertex, an edge and a face side that are mutually incident. An automorphism of a map is a permutation of its flags such that two flags sharing a vertex, edge or a face are sent to a pair of flags sharing the same type of elements. The collection of all such mappings under composition forms the automorphism group of a map, acting semi-regularly on the flag set. If this action is regular we say that the map is regular as well. Regular maps may be viewed as having the ‘highest level of symmetry’ among all maps. A slightly weaker concept is that of an orientably-regular map, where only orientable surfaces and orientation-preserving automorphisms are considered.

Regular maps embody three varieties of objects: a graph, a surface on which the graph is embedded, and a group isomorphic to the automorphism group of the embedding. Classification of regular and orientably-regular maps, which is a central problem in the theory of symmetric maps, is therefore usually attempted for a particular collection of underlying graphs, or carrier surfaces, or automorphism groups.

In the talk we will focus on classification of regular and orientably-regular maps with a given (isomorphism type of) automorphism group. We will give a survey of available results; as it turns out, two-dimensional linear groups
PSL(2,q) and PGL2,q) appear to be best understood from this point of view. Interestingly, the PGL(2,q) groups are one of the two infinite families of finite sharply 3-transitive permutation groups. Their ‘counterpart’, however, i.e., the ‘other’ such infinite family consisting of the so-called twisted linear groups, have been somewhat neglected in the literature. We will also present new results on enumeration of regular and orientably-regular maps with twisted linear automorphism groups and discuss open questions in this area of research.

15:-16:00 Break

16:00-17:00 William Norledge (Newcastle) Covering Theory of Combinatorial Buildings

Buildings are certain simply connected and highly symmetrical polyhedral complexes, which have a combinatorial description as a “Weyl-metric space”. Towards a greater understanding of locally compact groups, the groups which act on buildings are studied in a hope to extend the work of Bruhat-Tits on algebraic groups over non-Archimedean local fields acting on Euclidean buildings.
Covering theory of complexes of groups has been used to construct actions of groups on buildings. During the talk, we’ll introduce covering theory of buildings which doesn’t resort to complexes of groups, but rather deals directly with the Weyl-metric. We’ll use this covering theory to construct actions of amalgams of surface groups on hyperbolic buildings.