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 ≤ 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|.

Base sizes of primitive groups: Bounds with explicit constants

“…, k − 1}} is R-invariant. Since R is a subgroup of a transitive group on kℓ points which has order at most |G|, we see, by [13,Theorem 1.2], that d(R) ≤ 48 kℓ |G|.…”

“…For any integers j and u with 0 ≤ j ≤ k − 1 and 1 < u ≤ ℓ color jℓ + u with the color (α, β) where α is the color of jℓ + 1 in P and β is the color of jℓ + u in P. Clearly, no non-identity element of R preserves this new coloring. For ℓ ≥ 3, we see, by Lemma 2.1 of [13], that G has a base B containing…”

“…By using the idea of [12, Lemma 3.8] combined with Lemma 2.1 of [13], we see that The set Γ can be thought of as the set of right cosets in H of the subgroup H 0 = (D×Q)∩H where D denotes the diagonal subgroup of Aut(S) ℓ . In particular, |Γ| = |S| ℓ−1 .…”

“…Hence log |H| log |V | ≥ So assume now that b(K 1 ) ≥ 16 and b(K 1 ) ≤ 2(log |K 1 |/ log |V 1 |) + 9. In that case our proof strictly follows the arguments of [13], but in order to be able to prove Theorem 5.2 we need to give more precise estimates for the constants appearing there. In the proof we will freely use the concepts and notation of [13].…”

“…Despite a great deal of attention, Pyber's conjecture remained open until very recently, when it was proved in [13]. It is shown in [13] that there exists a universal constant c > 0 such that for every primitive permutation group G of degree n we have b(G) < 45(log |G|/ log n) + c.…”