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 $~$n$~$ there is a group $~$S_n$~$, the symmetric group of order $~$n$~$, defined as the group of all permutations (bijections) $~$\{ 1, 2, \dots n \} \to \{ 1, 2, \dots n \}$~$ (or any other Set with $~$n$~$ elements). The symmetric groups play a central role in group theory: for example, a group action of a group $~$G$~$ on a set $~$X$~$ with $~$n$~$ elements is the same as a homomorphism $~$G \to S_n$~$.

Up to conjugacy, a permutation is determined by its cycle type.

The dihedral groups

The dihedral groups $~$D_{2n}$~$ are the collections of symmetries of an $~$n$~$-sided regular polygon. It has a presentation $~$\langle r, f \mid r^n, f^2, (rf)^2 \rangle$~$, where $~$r$~$ represents rotation by $~$\tau/n$~$ degrees, and $~$f$~$ represents reflection.

For $~$n > 2$~$, the dihedral groups are non-commutative.

The general linear groups

For every field $~$K$~$ and positive integer $~$n$~$ there is a group $~$GL_n(K)$~$, the [general_linear_group general linear group] of order $~$n$~$ over $~$K$~$. Concretely, this is the group of all invertible $~$n \times n$~$ [matrix matrices] with entries in $~$K$~$; more abstractly, this is the [automorphism automorphism group] of a vector space of [vector_space_dimension dimension] $~$n$~$ over $~$K$~$.

If $~$K$~$ is [algebraically_closed_field algebraically closed], then up to conjugacy, a matrix is determined by its [Jordan_normal_form Jordan normal form].