Group: Examples

https://arbital.com/p/group_examples

by Qiaochu Yuan May 25 2016 updated Oct 21 2016

Why would anyone have invented groups, anyway? What were the historically motivating examples, and what examples are important today?


[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].