The symmetric group \(S_n\) consists of all permutations of \(n\) objects. It contains every degree-\(n\) permutation group and is the universal target of Cayley representations. [@isaacs2008]
Definition
A permutation is written as a product of disjoint cycles. Its cycle type is a partition of \(n\) and determines its conjugacy class in \(S_n\).
Core result
Every cycle is a product of transpositions, and adjacent transpositions generate every transposition. The normal-subgroup result uses simplicity of \(A_n\) and intersection with \(A_n\); see Theorems 8.27-8.28.
Structural properties
- \(|S_n|=n!\) and the natural action is \(n\)-transitive.
- The center is trivial for \(n\geq3\).
- In the standard nontrivial range, \(S_n'=A_n\).
- Conjugacy classes are cycle types, with centralizer order computed from cycle lengths and multiplicities.
- \(S_6\) has an exceptional outer automorphism, the unique higher-degree exception.
Symmetric groups serve simultaneously as containers for permutation groups, basic nonabelian examples, and extensions of alternating simple groups.
Example and boundary
Knowledge network
The alternating group is the index-two kernel, conjugacy classes are organized by cycle type, and multiple transitivity describes the natural action.
Proof and sources
Permutation foundations are in the appendix and Section 8A; simplicity and normal subgroups are Theorems 8.27-8.28. [@isaacs2008]
