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 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.