The orthogonal group, denoted as O(n), is the set of all n x n orthogonal matrices, which are matrices whose columns and rows are orthonormal vectors. This group is significant in the study of isometry groups, as it represents transformations that preserve the inner product, thus maintaining distances and angles in Euclidean spaces. Orthogonal transformations are crucial for understanding homogeneous spaces, as they allow for the exploration of symmetries in geometric structures.
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