LAUSR

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 ≤ b ( G ) 45 ( log ⁡ | G | / log ⁡ n ) + c . This finishes the proof of Pyber's base size conjecture. The main part of our paper is to prove this statement for affine permutation groups G = V ⋊ H where H ≤ G L ( V ) is an imprimitive linear group. 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 | G | n d ( G ) ≤ 48 | G | n .