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 $⟨r,f\mid r^n,f^2,(rf{)}^2⟩$, 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 $G{L}_n\left(K\right)$, the general linear group of order $n$ over $K$. Concretely, this is the group of all invertible $n\times n$ matrices with entries in $K$; more abstractly, this is the automorphism group of a vector space of dimension $n$ over $K$.

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