orthogonal group

The orthogonal group collects every linear motion of a space that leaves the form untouched — the symmetries of length and angle. For the ordinary dot product on R^n these are the rotations and reflections, the rigid motions fixing the origin. For a general quadratic form they are the linear maps that preserve its values, the ‘isometries’ of that geometry.

Given a quadratic space (V, Q), the orthogonal group O(Q) is the set of invertible linear maps g : V → V with Q(g v) = Q(v) for all v, equivalently B(g u, g v) = B(u, v) for the polar form. In coordinates, where Q has symmetric matrix A, this is the matrix group { g : g^T A g = A }. It is a group under composition: identity preserves Q, the composite of two isometries is an isometry, and inverses of isometries are isometries.

When A = I this is the classical O(n) of real orthogonal matrices, with O(n) = { g : g^T g = I }; the subgroup of determinant 1 is the special orthogonal group SO(n), the rotations. For indefinite real forms one gets groups like O(p, q), e.g. the Lorentz group O(3, 1). Over a finite field these become finite groups of Lie type, and many simple groups arise as their derived subgroups, tying quadratic forms to the classification of finite simple groups.