Algebra 1 by Ramji Lal
Author:Ramji Lal
Language: eng
Format: epub
Publisher: Springer Singapore, Singapore
Different choices of nonsingular symmetric (skew symmetric) matrices P give rise to different groups O(P) which play crucial roles in the respective geometries.
Orthogonal Groups
Orthogonal groups play a very crucial role in Euclidean and spherical geometries. Indeed, they are the group of isometries of the Euclidean and spherical spaces.
Consider the symmetric matrix . The group is denoted by O(n). Thus, . This group is called the orthogonal group . The members of O(n) are called the orthogonal matrices. If A is an orthogonal matrix, then . Thus, det induces a homomorphism from O(n) to the two element group . The diagonal matrix is an orthogonal matrix with determinant . Hence, det is a surjective homomorphism from O(n) to . The kenel of this homomorphism is the normal subgroup of O(n). This subgroup is denoted by SO(n), and it is called the special orthogonal group (also called the rotation group). From the fundamental theorem of homomorphism, we get the following corollary:
Corollary 6.4.6
O(n) / SO(n) is isomorphic to .
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