Hülya Duyan, Zoltán Halasi, Károly Podoski
August 31, 2019

### Abstract

The minimal base size b(G){b(G)} for a permutation group G is a widely studied topic in permutation group theory. Z. Halasi and K. Podoski [Every coprime linear group admits a base of size two, Trans. Amer. Math. Soc. 368 2016, 8, 5857–5887] proved that b(G)≤2{b(G)\leq 2} for coprime linear groups. Motivated by this result and the probabilistic method used by T. Burness, M. W. Liebeck and A. Shalev, it was asked by L. Pyber [Personal communication, Bielefeld, 2017] whether or not, for coprime linear groups G≤GL(V){G\leq GL(V)}, there exists a constant c such that the probability that a random c -tuple is a base for G tends to 1 as |V|→∞{\lvert V\rvert\to\infty}. While the answer to this question is negative in general, it is positive under the additional assumption that G is primitive as a linear group. In this paper, we show that almost all 11-tuples are bases for coprime primitive linear groups.