The Symmetric Group “An Example of Finite Nonabelian Group”

Abstract

This chapter discusses the group $${\mathfrak{S}}_{n}$$ , the symmetric group on $$n$$ elements, which is one of the most important examples of finite groups and is widely used in applications to geometry and physics. The importance of symmetry groups in abstract algebra is due to the fact that for any finite group $$G$$ , there is a symmetric group $${\mathfrak{S}}_{n}$$ that contains a copy of $$G$$ . For each $$n \in {\mathbb{N}}$$ , the group $${\mathfrak{S}}_{n}$$ consists of all the bijective maps of $$\left\{ {1,2, \ldots ,n} \right\}$$ to itself, called permutations of $$\left\{ {1,2, \ldots ,n} \right\}$$ . These permutations are usually denoted by symbols such as $$\phi \;{\text{and}}\;\psi$$ . The identity permutation that corresponds to the identity map of $$\left\{ {1,2, \ldots ,n} \right\}$$ is denoted by $$e$$ . In this chapter, Sect. 6.1 provides a representation of the elements of $${\mathfrak{S}}_{n}$$ as matrices and specifies the order of $${\mathfrak{S}}_{n}$$ in terms of the integer $$n$$ . Additionally, the notion of pairwise disjoint permutations is discussed, and their commutativity is verified. In Sect. 6.2, cycles, a special case of permutations, are defined and studied. The main result of this section is Proposition 6.2.9, which states that any permutation can be written as a finite product of disjoint cycles. The proof of this proposition requires a study of orbits of a permutation which discussed in Sect. 6.3 and followed by the proof of Proposition 6.2.9. The last two sections of this chapter discuss methods for determining the order of permutations and classifying permutations as odd and even.