# 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.

Parents:

• Group

The algebraic structure that captures symmetry, relationships between transformations, and part of what multiplication and addition have in common.