On the base size and minimal degree of transitive groups

TL;DR

This paper investigates the relationship between the minimal degree and base size of transitive permutation groups, demonstrating that their product can be nearly quadratic in the degree, and identifies classes with smaller bounds.

Contribution

It establishes the existence of transitive groups with nearly quadratic product of minimal degree and base size, and characterizes classes with smaller bounds similar to primitive groups.

Findings

Existence of transitive groups with  product  degree^{2-\u03b5}

Identification of classes with smaller bounds for base size and minimal degree

Comparison of bounds with primitive groups

Abstract

Let $G$ be a permutation group, and denote with $\mu(G)$ and $b(G)$ its minimal degree and base size respectively. We show that for every $\varepsilon>0$, there exists a transitive permutation group $G$ of degree $n$ with \[ \mu(G)b(G) \geq n^{2-\varepsilon}. \] We also identify some classes of transitive and intransitive groups whose base size and minimal degree have a smaller upper bound, shared with primitive groups.