Applications and adaptations of the low index subgroups procedure

Abstract

The low-index subgroups procedure is an algorithm for finding all subgroups of up to a given index in a finitely presented group G and hence for determining all transitive permutation representations of G of small degree. A number of significant applications of this algorithm are discussed, in particular to the construction of graphs and surfaces with large automorphism groups. Furthermore, three useful adaptations of the procedure are described, along with parallelisation of the algorithm. In particular, one adaptation finds all complements of a given finite subgroup (in certain contexts), and another finds all normal subgroups of small index in the group G. Significant recent applications of these are also described in some detail.