Base Sizes of Permutation Groups

Base Sizes of Permutation Groups

Mathematics

Eleanor Milburn

2025

This project report explores the base sizes of group actions. We begin with some preliminary definitions and results. Next, we investigate a combinatorial approach to proving the bound $|G| \geq 2^{b(G)}$; we give a conjecture for a stronger statement concerning a system of distinct representatives for sets $\left\{ \bigcap\limits_{\alpha \in \Omega}\mathrm{Stab}_G(\alpha) : \emptyset \neq \Omega \subseteq B \right\}$, where $B$ is a minimal base for a permutation group $G$. Subsequently, we follow a paper of Burness, O'Brien and Wilson in explaining various computational and character-theoretic techniques that can be used to compute base sizes. Finally, we apply these methods to a selection of almost simple sporadic groups and their subgroups.

Mathematics, algebra, group theory, base sizes