Algorithms for Finite Groups

Exercise 3.6. (a) Those congruences (norm-preserving linear transformations) of R which fix the origin form the orthogonal group O(n) consisting of the orthogonal matrices A defined by the equation AA = I. (b) Prove that the determinant of such a matrix is ±1. (c) The orientation preserving (“sense-presrving”) transformations form the group SO(n) which consists of those A ∈ O(n) with det(A) = 1. Prove: SO(n) is the unique subgroup of index 2 in O(n).