This chapter has introduced the stabiliser chain, the inductive foundation of the powerful algorithms for handling large permutation groups. The representation of elements by their base image, and as a product of coset representatives is central to the power of these algorithms. They are applied here to the performance of the tasks of testing membership, writing elements as products of the strong generators, and the enumeration of all the elements of a group.
We have presented several examples. It is hoped the reader will use them in the study of these and later algorithms.
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