# 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 $n$ 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].