Permutation groups, simple groups, and sieve methods (Q814164)

Let \(I\) be the set of indices of non-trivial subgroups of finite non-abelian simple groups except that of the alternating group on \(n-1\) letters in the alternating group in \(n\) letters. In the paper under review for any real number \(x>1\) it is shown that the number of integers \(n\) in \(I\) not exceeding \(x\) is ~\(hx/\log(x)\) for some given constant \(h\). As a consequence it is shown that for most positive integers \(n\) the only quasiprimitive permutation groups of degree \(n\) are the symmetric and the alternating groups on \(n\) letters. The proof is a combination of group theoretic and number theoretic methods.