Permutation groups and transformation semigroups : results and problems
Abstract
J.M. Howie, the influential St Andrews semigroupist, claimed that we value an area of pure mathematics to the extent that (a) it gives rise to arguments that are deep and elegant, and (b) it has interesting interconnections with other parts of pure mathematics. This paper surveys some recent results on the transformation semigroup generated by a permutation group G and a single non-permutation a. Our particular concern is the influence that properties of G (related to homogeneity, transitivity and primitivity) have on the structure of the semigroup. In the first part of the paper, we consider properties of S=<G,a> such as regularity and generation. The second is a brief report on the synchronization project, which aims to decide in what circumstances S contains an element of rank 1. The paper closes with a list of open problems on permutation groups and linear groups, and some comments about the impact on semigroups are provided. These two research directions outlined above lead to very interesting and challenging problems on primitive permutation groups whose solutions require combining results from several different areas of mathematics, certainly fulfilling both of Howie's elegance and value tests in a new and fascinating way.
Citation
Araujo , J & Cameron , P J 2015 , Permutation groups and transformation semigroups : results and problems . in C M Campbell , M R Quick , E F Robertson & C M Roney-Dougal (eds) , Groups St Andrews 2013 . London Mathematical Society Lecture Note Series , vol. 422 , Cambridge University Press , pp. 128-141 .
Publication
Groups St Andrews 2013
Status
Peer reviewed
ISSN
0076-0552Type
Book item
Rights
© 2015, Publisher / the Author(s). This work is made available online in accordance with the publisher’s policies. This is the author created, accepted version manuscript following peer review and may differ slightly from the final published version.
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