On the orders of doubly transitive permutation groups, elementary estimates (Q1803878)

Let \(G\) be a finite permutation group of degree \(n\) acting on the set \(\Omega\). A subset \(\Delta\) of \(\Omega\) is called a base of \(G\) iff the pointwise stabilizer of \(\Delta\) in \(G\) is the identity; \(b(G)\) denotes the minimum size of any base of \(G\). Clearly \(| G| \leq n^{b(G)}\). Using a short argument based on ideas of L. Babai it is proved that \(b(G) \leq c\log^ 2n\) for some constant \(c> 0\) if \(G\) is doubly transitive of degree \(n\) not containing the alternating group \(A_ n\) (Theorem A). \textit{L. Babai} had proved that \(b(G) \leq 4\sqrt{n}\ln n\) for uniprimitive \(G\) [Ann. Math., II. Ser. 113, 553-568 (1981; Zbl 0485.20002)]. The proofs given in the paper are elementary; the classification of finite simple groups (or of finite doubly transitive permutation groups) is not used. Also a second slightly weaker bound (Theorem B) is obtained which can be derived by a very short and fully self-contained combinatorial proof.