The orthogonal group, denoted as O(n), consists of all n x n real matrices that preserve the length of vectors when multiplied. This means that if you take a vector and multiply it by a matrix from the orthogonal group, the length of the resulting vector remains the same. The defining property of these matrices is that their transpose is equal to their inverse, which can be expressed mathematically as A^T A = I, where A is a matrix in O(n) and I is the identity matrix. Orthogonal groups are important in various fields, including linear algebra, geometry, and physics. They describe symmetries and rotations in n-dimensional space without distortion. For example, in 3D space, the orthogonal group captures all possible rotations and reflections, making it essential for understanding concepts like rigid body motion and conservation laws in physics.