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Seminario del 2021
2021
14 settembre
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.