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:

• 2.1 Permutations in RepLAB
• 2.2 Groups from scratch
• 2.2b Groups from graphs
• 2.3 Computing group properties
• 2.4 Subgroups
• (2.5 Conjugacy classes)
• (2.6 Cosets)