ABSTRACT
In Chapter 29 we introduced the symmetric groups Sn. These groups are of theoretical importance because every finite group is isomorphic to a subgroup of such a group, as Cayley’s Theorem 29.1 shows. They also provide us with many examples of non-abelian groups. In this chapter we inquire a bit more into permutations and introduce an efficient and illuminating notation for them. We first make a few notational remarks about permutation groups. Note that if m < n, we can think of Sm as a subgroup of Sn because
any permutation of {1, 2, · · · ,m} can be thought of as a permutation of the larger set {1, 2, · · · , n}, which leaves the elements m + 1,m + 2, · · · , n fixed. Technically speaking, we should use the language of isomorphism to describe this situation, but we will be content to be a little sloppy here and identify Sm as a subgroup of Sn. (See Exercise 29.2.) Consequently, in what follows we will not bother to be too specific about which permutation group a given permutation belongs to. Recall that the group operation in Sn is functional composition, since
the elements of Sn are actually functions from the set {1, 2, . . . , n} to itself. However, we will for the most part in this (and future chapters) speak less formally as group theorists about this operation; since it is written multiplicatively, we will tend to speak of the product of two permutations.
