A permutation is defined as a bijection from a set to itself.
For a set of size \(n\), the number of permutations is \(n!\). This is because there are \(n\) possibilities for the first item, \(n-1\) for the second and so on.
The set of all permutations forms a group, the symmetric group. This forms a group because:
There is an identity element
Each combination of permutations is also in the group.
Each permutation has an inverse in the group.
A subgroup of the symmetric group is called a permutation group.
A commutative group, that is where \(a\odot b=b\odot a\).
The following are abelian groups:
Integers and addition
\(\{-1, 1\}\) and multiplication
The following are not abelian groups:
\(n\times n\) matrices with determinants other than \(0\)
The group commutator is:
\([a,b]=a^{-1}b^{-1}ab\)
If the group is abelian then \([a,b]=0\). The group commutator is a measure of how non-abelian the group is.
This has the following properties:
Alternativity: \([A,A]=I\)
If we have two groups \(G\) and \(H\) we can form new group \(G\times H\).
For every \(g\in G\) and \(h\in H\) there is \((g,h)\in G\times H\).
The binary operation we have is:
\((g_1, h_1)(g_2,h_2)=(g_1g_2,h_1,h_2)\)