Mariapia Moscatiello (Università di Bologna)

Let G be a permutation group acting on a finite set Omega. A subset B of Omega is called a base for G if the pointwise stabilizer of B in G is trivial. In the 19th century, bounding the order of a finite primitive permutation group G was a problem that attracted a lot of attention. Early investigations of bases then arose because such a problem reduces to that of bounding the minimal size of a base of G. Some other far- reaching applications across Pure Mathematics led the study of the base size to be a crucial area of current research in permutation groups. In the first part of the talk, we will investigate some of these applications and review some results about base size. We will present a recent improvement of a famous estimation due to Liebeck that estimates the base size of a primitive permutation group in terms of its degree. In the second part of the talk, we will define the concept of irredundant bases of G and the concept of IBIS groups. Whereas bases of minimal size have been well studied, irredundant bases and IBIS groups have not yet received a similar degree of attention. Indeed, Cameron and Fon-Der-Flaas, already in 1995, defined such groups and proposed to classify some meaningful families. But only this year, a systematic investigation of primitive permutation IBIS groups has been started. We will discuss how we reduced the classification of primitive IBIS groups to the almost simple groups and affine groups. Eventually, we will conclude by mentioning recent advances towards a complete classification.