This chapter has considered the tasks of computing images and preimages of elements and subgroups under homomorphisms. In particular, it deals with the task of computing the kernel of the homomorphism. Most attention is given to the natural homomorphisms of a permutation group that arise from invariant subsets and partitions of the points, but the chapter also considers general homomorphisms defined in terms of the images of the group generators.
The algorithms determine a base and strong generating set of the subgroups constructed as kernels, images, or preimages. They are efficient, and the most expensive component in their execution is the base change algorithm.
Homomorphisms play an important role in divide- and-conquer algorithms for permutation groups, as the reader will see in the following chapters.
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