University of Bristol, Wednesday June 1st, 2016
School of Mathematics, University Walk, lectures will take place in room SM1.
After the last talk we will go for drinks and dinner.
Suppose that G is a transitive permutation group, of degree n, but that G needs a large number of generators (in terms of n). If possible, we would like to “reduce” the number of generators, whilst keeping our group transitive. More precisely, we would like to take a subset X of G, minimal with the property that <X> is transitive. The question is: can we find a good upper bound for |X|, in terms of n? In this talk, we discuss the history of this question, including an old conjecture of Pyber, and some new results. We will also briefly describe some of the applications of the question.
In recent years it has been noticed that knowledge about permutation groups can be applied to many old problems about transformation semigroups, especially considering regularity and idempotent generation. I have been working on this with Joao Araujo (Lisbon) and others. I will talk about two aspects of this.
A common type of problem asks what properties of a permutation group G force the semigroup generated by G and a non-permutation $a$ to have a particular property, for all or some specified elements a. Recently, we have considered the question of which permutation groups G have the property that, for any rank 2 map a, the semigroup <G,a> is idempotent generated. This condition forces G to be primitive, and is equivalent to the following property of G: for any orbit O of G on 2-sets, and any block B of imprimitivity for G acting on O, the graph with edge set O\B is connected. Such a group must be basic. It fails if G has an imprimitive subgroup of index 2, and it also fails for a class of primitive groups arising from triality. We conjecture, backed by substantial computational evidence, that it is true for all other primitive groups.
The other part of my talk will be an update on the ongoing synchronization project, which asks: which permutation groups G have the property that, for any non-permutation a, the semigroup <G,a> contains an element of rank 1.
For every doubly transitive permutation group G the conjugates
of a non-trivial coset of a point stabilizer H cover the group.
This implies that every non-trivial conjugacy class of elements
of G that contains an element of H does contain a transversal
of H in G and that every other non-trivial conjugacy class
contains a transversal for the set of cosets of H in G different
We study the finite groups satisfying this property; more precisely
the class of primitive permutation groups, called
CCI groups, that in fact properly contains the class of 2-transitive