Base sizes of primitive permutation groups

Abstract Let G be a permutation group, acting on set $$\varOmega $$ ? of size n . A subset $${\mathcal {B}}$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">B is base for if the pointwise stabilizer $$G_{({\mathcal {B}})}$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">G(B) trivial. b ( ) minimal subgroup $$\mathrm {Sym}(n)$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">Sym(n) large there exist integers m and $$r \ge 1$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">r?1 such that $${{\,\mathrm{Alt}\,}}(m)^r \unlhd \le {{\,\mathrm{Sym}\,}}(m)\wr {{\,\mathrm{Sym}\,}}(r)$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">Alt(m)r?G?Sym(m)?(r) , where action $${{\,\mathrm{Sym}\,}}(m)$$ />(m) k -element subsets $$\{1,\dots ,m\}$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">{1,?,m} wreath product acts with action. In this paper we prove primitive not base, then either Mathieu group {M}_{24}$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">M24 in its natural 24 points, or $$b(G)\le \lceil \log n\rceil +1$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">b(G)??logn?+1 Furthermore, show are infinitely many groups which $$b(G) > + xmlns:mml="http://www.w3.org/1998/Math/MathML">b(G)>logn+1 so our bound optimal.