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.

### 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.

** 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.

### 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.

### 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.)

### 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.

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### 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.

### On the derangement graph of acting on the projective line

* Pablo Spiga *(UWA), 3 November 2009

Given a finite set and a family of -subsets of , we say that is intersecting if any two elements in have non-empty intersection. A classical theorem in extremal combinatorics of Erdos-Ko-Rado classifies the independent sets of maximal size for . 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 acting on the projective line.

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