Base sizes of primitive groups: bounds with explicit constants

TL;DR

This paper establishes explicit bounds on the minimal base size of finite primitive permutation groups, showing it is closely related to the group's order and degree, with bounds that are tight and asymptotically optimal.

Contribution

The paper provides the first explicit bounds with constants for the minimal base size of primitive groups, improving understanding of their structure and limitations.

Findings

Bound on base size: at most 2 (log |G| / log n) + 24.

Existence of groups reaching the bound asymptotically.

Primitive groups not containing Alt(n) have bases of size at most max{√n, 25}.

Abstract

We show that the minimal base size $b(G)$ of a finite primitive permutation group $G$ of degree $n$ is at most $2 (\log |G|/\log n) + 24$. This bound is asymptotically best possible since there exists a sequence of primitive permutation groups $G$ of degrees $n$ such that $b(G) = \lfloor 2 (\log |G|/\log n) \rceil - 2$ and $b(G)$ is unbounded. As a corollary we show that a primitive permutation group of degree $n$ that does not contain the alternating group $\mathrm{Alt}(n)$ has a base of size at most $\max\{\sqrt{n} , \ 25\}$.