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The orthogonal group is an algebraic group and a Lie group. It is compact. The orthogonal group in dimension n has two connected components. The one that contains the identity element is a normal subgroup, called the special orthogonal group, and denoted SO(n). It consists of all orthogonal matrices of determinant 1.
It is a subgroup of the orthogonal group O(3), the group of all isometries that leave the origin fixed, or correspondingly, the group of orthogonal matrices. O(3) itself is a subgroup of the Euclidean group E(3) of all isometries. Symmetry groups of geometric objects are isometry groups.
Every such group is a subgroup of the orthogonal group O(2), including O(2) itself. Its elements are rotations and reflections, and every such group containing only rotations is a subgroup of the special orthogonal group SO(2), including SO(2) itself.
The indefinite special orthogonal group, SO(p, q) is the subgroup of O(p, q) consisting of all elements with determinant 1. Unlike in the definite case, SO( p , q ) is not connected – it has 2 components – and there are two additional finite index subgroups, namely the connected SO + ( p , q ) and O + ( p , q ) , which has 2 components ...
In geometry, a point group is a mathematical group of symmetry operations (isometries in a Euclidean space) that have a fixed point in common. The coordinate origin of the Euclidean space is conventionally taken to be a fixed point, and every point group in dimension d is then a subgroup of the orthogonal group O(d).
The group depends only on the dimension n of the space, and is commonly denoted E(n) or ISO(n), for inhomogeneous special orthogonal group. The Euclidean group E( n ) comprises all translations , rotations , and reflections of E n {\displaystyle \mathbb {E} ^{n}} ; and arbitrary finite combinations of them.
For an orthogonal matrix R, note that det R T = det R implies (det R) 2 = 1, so that det R = ±1. The subgroup of orthogonal matrices with determinant +1 is called the special orthogonal group, denoted SO(3). Thus every rotation can be represented uniquely by an orthogonal matrix with unit determinant.
Likewise, the orthogonal group O(q) is the subgroup of the general linear group that preserves a nondegenerate quadratic form q on a vector space over a field k. The algebraic group O ( q ) has two connected components , and its identity component SO ( q ) is reductive, in fact simple for q of dimension n at least 3.