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 ≤ GL(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 n |G| < d(G) ≤ 48 n |G|.

The minimal base size b(G) for a permutation group G, is a widely studied topic in the permutation group theory. Z. Halasi and K. Podoski [8] proved that b(G)

No abstract