first in the K-orbit test in search,
first in the Kγ1, γ2, ..., γi−1, -orbit test in generate, and
weak test for first in the right coset of K.
Of these, the first is of great significance, leading to great improvements in the backtrack search.
restricting the choice of images using ¯ΩP,
choosing a base appropriate to the problem,
using any known subgroups, and
generating images of an initial base segment.
There is a close connection between the first two. Indeed, the effectiveness of (1) depends on (2), and the selection of a base depends on how it will be used in (1). Both offer significant improvements in the backtrack searches that have been discussed. The importance of (3) and (4) depends on the problem.
The algorithms for the construction of centralizers, normalizers, intersections, and set stabilisers have been presented. The techniques of this chapter are also used in the construction of the automorphism group of a group.
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