Applications and Adaptations of the Low Index Subgroups Process

The low index subgroups process may be used to nd subgroups of small index in a nitely-presented group G, and hence to determine all transitive permutation representations of G of small degree. A number of successful applications will be discussed, in particular to the construction of graphs and surfaces with large automorphism groups. Further, two useful adaptations of the process will be described: one which nds all complements of a given subgroup of nite index, and one which nds all normal subgroups of small index in the group G. Signiicant applications of these will also be mentioned. Extended abstract Given a nitely-presented group G = hX j Ri, where X is a nite set of generators and R is a nite set of relators (each expressed as a word on X X ?1), there is an algorithm due to Sims which allows the determination of a representative from each conjugacy class of subgroups of G of index up to some speciied integer N. Known as the \low index subgroups process" and described in detail in 7], this is a powerful algorithm with several uses and important applications. For one thing, obviously it provides a lot of information about the structure of the group G, \from the top down". Together with the Reidemeister-Schreier process 9] or cohomological methods 10], it can often be used to determine whether or not G is innnite, or at least to obtain a lower bound on the order of G. More generally, since every subgroup H of nite index n in G corresponds to a transitive permutation representation of G on n points (equivalent to the right representation of G on cosets of H), it can be used to determine all nite factor groups of G isomorphic to permutation groups of small degree. The latter observation has been exploited in a number of areas where nitely-presented groups play an important role. For example, the automorphism group of every trivalent symmetric graph is a factor group of one of seven nitely-presented groups, and the low index subgroups process was used to nd interesting cases of such factor groups and the associated graphs in 4]. In turn the discovery of these examples motivated the construction 1