On the minimal degree and base size of finite primitive groups

TL;DR

This paper establishes a new upper bound on the base size of finite primitive groups and proves a tight relationship between minimal degree and base size, enhancing understanding of their structural properties.

Contribution

It introduces a new upper bound for the base size of primitive groups and relates minimal degree and base size with a nearly optimal inequality.

Findings

For primitive groups, the base size is bounded above by a function of the degree.

The product of minimal degree and base size is at most proportional to n log n for most primitive groups.

The bound is tight up to a constant factor, except for the Mathieu group of degree 24.

Abstract

Let $G$ be a finite permutation group acting on $\Omega$. A base for $G$ is a subset $B \subseteq \Omega$ such that the pointwise stabilizer $G_{(B)}$ is the identity. The base size of $G$, denoted by $b(G)$, is the cardinality of the smallest possible base. The minimal degree of $G$, denoted by $\mu(G)$, is the smallest cardinality of the support of a non trivial element of $G$. In this paper, we establish a new upper bound for $b(G)$ when $G$ is primitive, and subsequently prove that if $G$ is a primitive group different from the Mathieu group of degree $24$, then $\mu(G)b(G)\leq n \log n$, where $n$ is the degree of $G$. This bound is best possible, up to a multiplicative constant.