# Symmetric groups (S_n)

## Creating groups

### Permutations and the
symmetric group

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 symmetric group

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.

#### Permutation groups

A subgroup of the symmetric group is called a permutation group.

## Abelian groups

### Abelian groups

A commutative group, that is where \(a\odot
b=b\odot a\).

The following are abelian groups:

The following are not abelian groups:

### The group commutator

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:

## Direct product

### The direct product of groups

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:

\((g_1,
h_1)(g_2,h_2)=(g_1g_2,h_1,h_2)\)