New postdoc position

Cheryl Praeger and Stephen Galsby are currently advertising a one year postdoc positon in our research group. More details and information about how to apply are available at

http://external.jobs.uwa.edu.au/cw/en/job/499094/

 

The closing date is September 1st.

 

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Forrest Fellowships

The Forrest Foundation are currently advertising their Forrest Fellowships which are 3 year postdocs at a university in Western Australia.  Of course, The University of Western Australia is one such university.

They are looking for `outstanding researchers of exceptional ability and resourcefulness, having the highest calibre of academic achievements and with the potential to make a difference in the world.’ All research disciplines are eligible to apply. Applicants need to be no more than 2 years post PhD graduation.

More details can be found here. Note that one of the selection criteria involves ability and commitment to mentoring. Applications close at the end of June.

Applicants will need a statement of support from the univeristy that we wish to be at. If you are interested in applying for one of these to join the CMSC then feel free to contact one of us.

8th Slovenian Conference on Graph Theory

Last week I was at the 8th Slovenian Conference on Graph Theory.  This was the latest in what is commonly known as the ‘Bled conference’ but this year was in Kranjska Gora. This meant that the conference excursion was to Lake Bled. It was a very enjoyable conference with lots of interesting talks and it was good to catch up with lots of people. I was one of the plenary speakers and my talk was entitled ‘Bounding the number of automorphisms of a graph’. This surveyed the recent work on the Weiss conjecture and its generalisation the PSV conjecture. It also discussed my recent work with Luke Morgan on the PSV conjecture for semiprimitive groups with a nilpotent regular normal subgroup.  More details can be found on my slides.

Symmetries of Graphs and Networks IV

Last week I was at the Symmetries of Graphs and Networks IV conference at Rogla in Slovenia. The conference webpage is here.  At the same time was the annual  PhD summer school in discrete maths. As usual it was a very enjoyable and well organised conference. It was good to catch up with some of the regulars and meet a few new people as well

I was one of the invited speakers and spoke about some of the work that I have been doing recently with Luke Morgan on graph-restrictive permutation groups. The slides are available here.  The two relevant preprints are on the arxiv here and here.

 

 

 

 

Regular cycles of elements in finite primitive permutation groups

Cheryl Praeger, Pablo Spiga and I have just uploaded to the arxiv our preprint  Finite primitive permutation groups and regular cycles of their elements.   Everyone recalls from their first course in group theory that if you write a permutation in cycle notation, for example (1,2)(3,4,5)(6,7,8,9,10), then its order is the lowest common multiple of it cycle lengths. As I have just demonstrated, not every permutation must have a cycle whose length is the same as the order of the element. We call a cycle of an element g whose length is the order of g, a regular cycle.  A natural question to ask is when can you guarantee that a permutation has a regular cycle? Or in particular, for which permutation groups do all elements have a regular cycle?

Clearly, the full symmetric group contains elements with no regular cycles, but what about other groups?  Siemons and Zalesskii showed that for any group G  between PSL(n,q) and PGL(n,q) other than for (n,q)=(2,2) or (2,3), then in any action of G, every element of G has a regular cycle, except G=PSL(4,2) acting on  8 points. The exceptions are due to isomorphisms with the symmetric or alternating groups.  They also later showed that for any finite simple group G,other than the alternating group, that admits a 2-transitive action, in any action of G every element has a regular cycle. This was later extended by Emmett and Zalesskii to any finite simple classical group not isomorphic to PSL(n,q).

With these results in mind, we started investigating elements in primitive permutation groups. The results of this work are in the preprint. First we prove that for k\leq m/2, every element of Sym(m) in its action on k-sets has a regular cycle if and only if m is less than the sum of the first k+1 primes. After further computation we were then willing to make the following conjecture:

Conjecture: Let G\leqslant Sym(\Omega) be primitive such that some element has no regular cycle. Then there exist integers k\geq1, r\geq1 and m\geq5 such that
G preserves a product structure on \Omega=\Delta^r with |\Delta|=\binom{m}{k}, and Alt(m)^r  \vartriangleleft G\leqslant Sym(m)\textrm{wr} Sym(r), where Sym(m) induces its k-set action on \Delta.

The general philosophy behind why such a result should be true is that most primitive groups are known to be small in terms of a function of the degree.  There is a large body of work bounding the orders of primitive permutation groups with the best results due to Maroti. The actions of Sym(m) on k-sets are the usual exceptions.

Our paper  goes about attempting to prove this conjecture. We make substantial progress and reduced its proof to having to deal with all the primitive actions of classical groups. Note that Emmett and Zalesskii only dealt with simple ones.  One consequence of our work is we showed that every automorphism of a finite simple group has a regular cycle in its action on the simple group.

Pablo and Simon Guest have subsequently gone on to prove the result for all actions of classical groups and so the conjecture is now a theorem.

Pablo gave a great talk about the conjecture and its subsequent proof  at the recent BIRS workshop on Permutation Groups in Banff which you can view here.

Position at UWA

A position here in the School of Maths and Stats at UWA has just been advertised.  It is a full-time continuing position as either an Assistant or Associate Professor (Level B/C)  and the school is looking for ` a generalist mathematician with excellent skills in teaching and learning to develop and deliver a range of first year units to students from across the University.’  More details are available at the link above and queries should be directed to the Head of School, Andrew Bassom.

37 ACCMCC

This year’s Australasian Conference on Combinatorial Mathematics and Combinatorial Computing will be held here at UWA from the 9th to 13th of December. Put the date in your diary now and start looking for cheap flights to Perth.

The conference webpage is available here, where you can find the exciting lineup of invited speakers that we have lined up so far.

Details of previous conferences in the series are available at the CMSA webpage.

3rd SYGN Workshop

Last week I was at the 3rd Symmetries of Graphs and Networks Worskhop and PhD Summer School in Discrete Mathematics at Rogla, Slovenia.  The SYGN meeting has been held every two years with the first being at Banff in 2008 and this was the second one held at Rogla. The summer school part continues on the tradition started last year.

Once again the Slovenians did a great job at organising it and it was a very enjoyable conference. I was invited to give a minicourse as part of the summer school on The Polycirculant Conjecture. I have touched on this previously. Boštjan Kuzman kindly took  latex notes of my lectures and these are available at the conference webpage together with the slides from the other courses and talks from the workshop. The other courses were on Graph enumeration by Stephan Wagner and Finite geometries by György Kiss.

Four talks that I found particularly interesting were:

  • Roman Nedela’s survey on regular maps. This gave a great overview of the area and I learned a lot about what the main problems in the area are.
  • Jozef Siran’s survey on the degree-diameter problem.
  • Joy Morris’s talk on the structure of circulant graphs.
  • Steve Wilson’s talk on Tales from the census. This is his census with   Primož Potočnik on 4-valent edge-transitive graphs.

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