A proof of Pyber's base size conjecture

TL;DR

This paper proves Pyber's base size conjecture by establishing bounds on the minimal base size of primitive permutation groups in relation to their order and degree, confirming a long-standing mathematical hypothesis.

Contribution

The paper completes the proof of Pyber's base size conjecture, providing explicit bounds connecting group order, degree, and base size for primitive permutation groups.

Findings

Established bounds on the minimal base size of primitive groups

Proved the conjecture that relates base size to group order and degree

Derived estimates for the distinguishing number of transitive groups

Abstract

Building on earlier papers of several authors, we establish that there exists a universal constant $c > 0$ such that the minimal base size $b(G)$ of a primitive permutation group $G$ of degree $n$ satisfies $\log |G| / \log n \leq b(G) < 45 (\log |G| / \log n) + c$. This finishes the proof of Pyber's base size conjecture. An ingredient of the proof is that for the distinguishing number $d(G)$ (in the sense of Albertson and Collins) of a transitive permutation group $G$ of degree $n > 1$ we have the estimates $\sqrt[n]{|G|} < d(G) \leq 48 \sqrt[n]{|G|}$.