Subgroups of minimal index in polynomial time

By applying an old result of Y. Berkovich, we provide a polynomial-time algorithm for computing the minimal possible index of a proper subgroup of a finite permutation group [Formula: see text]. Moreover, we find that subgroup explicitly and within the same time if [Formula: see text] is given by a Cayley table. As a corollary, we get an algorithm for testing whether or not a finite permutation group acts on a tree non-trivially.

Number fields with large minimal index containing quadratic subfields

In this paper, we consider number fields containing quadratic subfields with minimal index that is large relative to the discriminant of the number field. We give new upper bounds on the minimal index, and construct families with the largest possible minimal index.