These groups are closely related not only is SO(2) a subgroup of O(.pdf
- 1. These groups are closely related: not only is SO(2) a subgroup of O(2) because any two reflections gives a rotation.Over the field R of real numbers, the orthogonal group O(n,?R) and the special orthogonal group SO(n,?R) are often simply denoted by O(n) and SO(n) if no confusion is possible. They form real compact Lie groups of dimension n(n ? 1)/2. O(n,?R) has two connected components, with SO(n,?R) being the identity component, i.e., the connected component containing the identity matrix. SO(n,?R) is a subgroup of E+(n), which consists of direct isometries, i.e., isometries preserving orientation; it contains those that leave the origin fixed Solution These groups are closely related: not only is SO(2) a subgroup of O(2) because any two reflections gives a rotation.Over the field R of real numbers, the orthogonal group O(n,?R) and the special orthogonal group SO(n,?R) are often simply denoted by O(n) and SO(n) if no confusion is possible. They form real compact Lie groups of dimension n(n ? 1)/2. O(n,?R) has two connected components, with SO(n,?R) being the identity component, i.e., the connected component containing the identity matrix. SO(n,?R) is a subgroup of E+(n), which consists of direct isometries, i.e., isometries preserving orientation; it contains those that leave the origin fixed