Padova Lecture Series 2021

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Bases for permutation groups

The notion of a base for a permutation group is a classical concept in group theory, which arises naturally in several different areas. The minimal cardinality of a base is called the base size of the group; this is an interesting invariant that has been widely studied, especially in the context of finite primitive groups.

In these lectures, which will be accessible to graduate students, we will begin by introducing the key concepts and we will recall some of the main results, both old and new. The Classification of Finite Simple Groups has revolutionised the study of bases, leading to several important advances in recent years. In the 1990s, for example, Pyber and Cameron posed highly influential conjectures concerning bases for primitive groups, which have finally been settled in the last few years. We will review some of the main ideas and techniques arising in this work, with a focus on the almost simple primitive groups. In particular, we will explain how powerful probabilistic methods have played an important role in some of these recent advances.

We will also look at recent applications of bases to a diverse range of problems, from the generation of finite groups to the classical notion of extreme primitivity. Finally, time permitting, we will discuss Jan Saxl's base-two project and we will introduce the Saxl graph of a permutation group, which arises naturally in this setting. This combinatorial object has some interesting properties -- we will review some of the main results and open problems.


The lectures