Abstract

A base of a permutation group $G$ acting on $\Omega$ is a subset of $\Omega$ whose pointwise stabiliser in $G$ is trivial. Bases have their origins in computational group theory, where they were used to efficiently store permutation groups of large degree into a small amount of computer memory. The minimal base size of $G$ is denoted by $b(G)$. A well-known conjecture made by Pyber in 1993 states that there is an absolute constant $c$ such that if $G$ acts primitively on $\Omega$, then $b(G) < c \log |G| / \log n$, where $| \Omega | = n$. After over 20 years and contributions by a variety of authors, Pyber’s conjecture was established in 2016 by Duyan, Halasi and Maroti. In this talk, I will cover some of the history and uses of bases, and discuss Pyber’s conjecture, especially in the context of primitive linear groups, and present some results on the determination of the constant $c$ for bases of almost quasisimple linear groups.