# 6: Groups Acting on Sets

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Until now we have thought of permutations mostly as ways of listing the elements of a set. In this chapter, we will find it very useful to think of permutations as functions. This will help us in using permutations to solve enumeration problems that cannot be solved by the quotient principle because they involve counting the blocks of a partition in which the blocks don’t have the same size. We begin by studying the kinds of permutations that arise in situations where we have used the quotient principle in the past.

• 6.1: Permutation Groups
Until now we have thought of permutations mostly as ways of listing the elements of a set. In this chapter we will find it very useful to think of permutations as functions. This will help us in using permutations to solve enumeration problems that cannot be solved by the quotient principle because they involve counting the blocks of a partition in which the blocks don’t have the same size.
• 6.2: Groups Acting on Sets
We have seen that the fact that we have defined a permutation group as the permutations of some specific set doesn’t preclude us from thinking of the elements of that group as permuting the elements of some other set as well.
• 6.3: Pólya-Redfield Enumeration Theory
George Pólya and Robert Redfield independently developed a theory of generating functions that describe the action of a group G on colorings of a set S by a set T when we know the action of G on S. Pólya’s work on the subject is very accessible in its exposition, and so the subject has become popularly known as Pólya theory, though Pólya-Redfield theory would be a better name. In this section we develop the elements of this theory.
• 6.4: Groups Acting on Sets (Exercises)
This section contains the supplementary problems related to the materials discussed in Chapter 6.