Orthogonal Group

The orthogonal group in dimension n, written O(n), is the group of distance preserving transformations of an n dimensional Euclidean space that fix the origin, with the group operation given by composing transformations. It is sometimes called the general orthogonal group, by analogy with the general linear group, and it is equivalently the group of n by n orthogonal matrices under matrix multiplication, an orthogonal matrix being a real matrix whose inverse equals its transpose. The orthogonal group in dimension n has two connected pieces: the one containing the identity is a normal subgroup called the special orthogonal group, written SO(n) and also known as the rotation group since its elements are exactly the rotations around the origin, consisting of every orthogonal matrix with determinant one; the other piece consists of the orthogonal matrices with determinant negative one, and does not itself form a group.