2. Permutation groups

The first symmetries we will describe are permutations acting on a finite set \(X\). Because \(X\) is finite, we can assume that \(X = \{1,2,\ldots,n\}\). We write \(S_n\) the set of all possible permutations of elements in the set. These permutations form a group:

  • The identity permutation sends \(1\) to \(1\), \(2\) to \(2\), … and \(n\) to \(n\).

  • Permutations compose, and this composition is associative.

  • The inverse permutation exists.

Here are the available topics on permutation groups on this subject: