Subgroups of minimal index in polynomial time

Abstract

Let G be a finite group and let H be a proper subgroup of G of minimal index. By applying an old result of Y. Berkovich, we provide a polynomial algorithm for computing |G : H| for a permutation group G. Moreover, we find H explicitly if G is given by a Cayley table. As a corollary, we get an algorithm for testing whether a finite permutation group acts on a tree or not.