A proof of Pyber's base size conjecture

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 $\log |G|/\log n\le b(G)<45(\log |G|/\log n)+c$. 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 $\sqrt[n]{|G|}<d(G)\le 48\sqrt[n]{|G|}$.

27 pages, referee comments included

References in corpus (3)

• The base size of a primitive diagonal group
• Structure and automorphisms of primitive coherent configurations
• Base sizes of imprimitive linear groups and orbits of general linear groups on spanning tuples

Cited by in corpus (7)

• Bases of twisted wreath products
• Normalisers of primitive permutation groups in quasipolynomial time
• Bases of quasisimple linear groups
• On the Burness-Giudici Conjecture
• Base sizes for primitive groups with soluble stabilisers
• On stabilizers in finite permutation groups
• The distinguishing number of quasiprimitive and semiprimitive groups