In this chapter, we collect some basic algorithms that serve as building blocks to higher level problems. Frequently, the efficient implementation of these low-level algorithms is the key to the practicality of the more glamorous, high-level problems that use them as subroutines.

Consequences of the Schreier–Sims Method

The major applications of the Schreier–Sims SGS construction are membership testing in groups and finding the order of groups, but an additional benefit of the algorithm is that its methodology can be applied to solve a host of other problems in basic permutation group manipulation. We list the most important applications in this section. The running times depend on which version of the Schreier–Sims algorithm we use; in particular, all tasks listed in this section can be performed by nearly linear-time Monte Carlo algorithms. For use in Chapter 6, we also point out that if we already have a nonredundant base and SGS for the input group then the algorithms presented in Sections 5.1.1–5.1.3 are all Las Vegas. Concerning the closure algorithms in Section 5.1.4, see Exercise 5.1.

Pointwise Stabilizers

Any version of the Schreier–Sims method presented in Chapter 4 can be easily modified to yield the pointwise stabilizer of some subset of the permutation domain. Suppose that G = 〈S〉 ≤ Sym(Ω) and Δ ⊆ Ω are given, and we need generators for G(Δ).