# Seminars

This page will collate the seminars given in the Groups, Combinatorics and Computation seminar.

### Three Hamilton Decomposition Problems

Brian Alspach(University of Newcastle) 11 May 2010

This talk deals with three middle-aged problems on decomposing graphs into Hamilton cycles. There will be something old, something new, something borrowed, and something blue.

Brian’s talk

### On arc-transitive groups with large arc-stabilisers

Gabriel Verret (University of Ljubljana) 16 March 2010

A classical result of Tutte is that the order of the arc-stabiliser in a cubic arc-transitive graph is bounded above by 16. This surprising result is both interesting from a theoretical point of view and it also has applications, for example in enumerating small cubic arc-transitive graphs. We wish to understand under what hypothesis Tutte’s result generalizes to other valencies and what can be said when it does not.

Gabriel’s talk

### The generalised Curtis-Tits system and black box groups

Şükrü Yalçinkaya (UWA)   9  March 2010.

The Curtis-Tits presentation of groups of Lie type is the main identification theorem used in the classification of the finite simple groups. I will describe the most general form of the Curtis-Tits presentation of finite groups of Lie type where the Phan’s presentation for the twisted groups appears as a special case. I will also talk about a beautiful application of this result to the recognition of finite black box groups.

If time permits I will briefly talk about how Curtis-Tits and its Phan variations link two theories: Theory of black box groups and groups of finite Morley rank.

Sukru’s talk

### A Solvable Version of the Baer-Suzuki Theorem

Simon Guest (UWA/Baylor University), 14 January 2010

Let G be a finite group, and take an element x in G. The Baer–Suzuki states that if every pair of conjugates of x generates a nilpotent group then the group generated by all of the conjugates of x is nilpotent. It is natural to ask if an analogous theorem is true for solvable groups. Namely, if every pair of conjugates of x generates a solvable group then is the group generate by all of the conjugates of x solvable? In fact, this is not true. For example, if x has order 2 in a (nonabelian) simple group G then every pair of conjugates of x generates a dihedral group (which is solvable), but the normal subgroup generated by all of the conjugates of x must be the whole of the nonabelian simple group G, which of course is not solvable. There are also counterexamples when x has order 3. However, the following is true:
(1) Let x in G have prime order p > 4. If every pair of conjugates of x generates a solvable group then the group generated by all of the conjugates of x is solvable.
(2) Let x in G be an element of any order. If every 4-tuple of conjugates x, x^{g_1}, x^{g_2}, x^{g_3} generates a solvable group then the group generated by all of the conjugates of x is solvable..

We will discuss these results, some generalizations, and some of the methods used in their proof.

Simon’s talk PDF

### Majorana representations of dihedral, alternating, and symmetric groups

Akos Seress (Ohio State University/UWA), 14 January 2010

### Buekenhout-Metz unitals

Nicola Durante (Università di Napoli “Federico II,”), 14 January 2010

We will discuss on some recent characterization theorems for Buekenhout-Metz unitals in a Desarguesian projective plane of square order.

### A geometric approach to Mathon maximal arcs

Frank De Clerck (Ghent University), 14 January 2010

A maximal arc of degree d in a projective plane of order q is a non-empty, proper subset of points such that every line meets the set in 0 or d points, for some d. If a plane has a maximal arc of degree d the dual plane has one of degree q/d. We will mainly restrict to Desarguesian planes. It has been proved by Ball, Blokhuis and Mazzocca that non-trivial maximal arcs in PG(2,q) can not exist if q is odd. They do exist if q is even: examples are hyperovals, Denniston arcs, Thas arcs and Mathon arcs. We will give an overview of these constructions and of the connection with other geometric topics. We will give a geometric approach to the Mathon arcs emphasising on those of degree 8.

### Quantum Geometry – MUB’s and SIC-POVM’s

Chris Godsil (Waterloo), 1 December 2009

Quantum physicists can use various interesting geometric structures in complex space for measurement, cryptographic protocols, etc. I will introduce these structures and show how they are related to classical objects in combinatorics and finite geometry. (No knowledge of physics will be needed.)

Chris’s talk PDF

### Analysing random graph processes using differential equations

Stephen Howe (ANU), 24 November 2009

Differential equations often arise when approximating discrete processes by continuous processes. For example, in the exponential model of population growth and radioactive decay. We may also consider a continuous approximation of a random graph process; again differential equations arise naturally. By solving these differential equations we can determine properties of the random process. I will give an introduction to this approach and describe some applications.

[No PDF]

### Combinatorics on Words and Some Applications in Number Theory

Amy Glen (Murdoch), 17 November 2009

I will briefly survey some old and new results concerning palindromic properties of infinite words and some applications to problems in Number Theory.

Amy’s talk PDF

### On the derangement graph of $PGL(2,q)$ acting on the projective line

Pablo Spiga (UWA), 3 November 2009

Given a finite set $X$ and a family $S$ of $k$-subsets of $X$, we say that $S$ is intersecting if any two elements in $S$ have non-empty intersection. A classical  theorem in extremal combinatorics of Erdos-Ko-Rado classifies the independent sets of maximal size for  $2k<|X|$. There are many applications  and generalizations  of this theorem in different areas of mathematics. The extension we are interested in deals with permutation groups. In particular, inspired by a recent paper of Godsil-Meagher, we prove an analogue of the Erdos-Ko-Rado theorem for the derangement graph of $PGL(2,q)$ acting on the projective line.

Pablo’s Talk PDF