[summary: Examples of groups, including the symmetric groups and [-general_linear_group general linear groups]. ]
The symmetric groups
For every positive integer there is a group , the symmetric group of order , defined as the group of all permutations (bijections) (or any other Set with elements). The symmetric groups play a central role in group theory: for example, a group action of a group on a set with elements is the same as a homomorphism .
Up to conjugacy, a permutation is determined by its cycle type.
The dihedral groups
The dihedral groups are the collections of symmetries of an -sided regular polygon. It has a presentation , where represents rotation by degrees, and represents reflection.
For , the dihedral groups are non-commutative.
The general linear groups
For every field and positive integer there is a group , the [general_linear_group general linear group] of order over . Concretely, this is the group of all invertible [matrix matrices] with entries in ; more abstractly, this is the [automorphism automorphism group] of a vector space of [vector_space_dimension dimension] over .
If is [algebraically_closed_field algebraically closed], then up to conjugacy, a matrix is determined by its [Jordan_normal_form Jordan normal form].