In this chapter, we develop a nearly linear-time library for constructing certain important subgroups of a given group. All algorithms are of the Monte Carlo type, since they are based on results of Section 4.5. However, if a base and SGS are known for the input group, then all algorithms in this chapter are of Las Vegas type (and in most cases there are even deterministic versions).

A current research project of great theoretical and practical interest is the upgrading of Monte Carlo permutation group algorithms to Las Vegas type. The claim we made in the previous paragraph implies that it is enough to upgrade the SGS constructions; we shall present a result in this direction in Section 8.3. To prove that all algorithms in this chapter are of Las Vegas type or deterministic, we always suppose that the input is an SGS S for some G ≤ Sym(Ω) relative to some base B, S satisfies S = S-1, and transversals corresponding to the point stabilizer chain defined by B are coded in shallow Schreier trees. Throughout this chapter, shallow Schreier tree means a Schreier tree of depth at most 2 log |G|. We remind the reader that, by Lemma 4.4.2, given an arbitrary SGS for G, a new SGS S satisfying S = S-1 and defining a shallow Schreier tree data structure can be computed in nearly linear time by a deterministic algorithm.